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\newtheorem{dfn}{Definition}[section]
\def\proof{\par\medskip\noindent{\it Proof: }}
\newtheorem{rem}[dfn]{Remark}
\newtheorem{thm}[dfn]{Theorem}
\newtheorem{lem}[dfn]{Lemma}
\newtheorem{claim}[dfn]{Claim}
\newtheorem{prop}[dfn]{Proposition}
\newtheorem{cor}[dfn]{Corollary}
\newtheorem{conj}[dfn]{Conjecture}
\newtheorem{ex}[dfn]{Example}
\newtheorem{exe}[dfn]{Exercise}
\newtheorem{defn}[dfn]{Definition}
\newtheorem{ques}[dfn]{Question}
\def\C{\Bbb C}
\def\R{\Bbb R}
\def\H{\Bbb H}
\def\Z{\Bbb Z}
\def\Q{\Bbb Q}
\def\E{\Bbb E}
\def\S{\Bbb S}
\def\B{\Bbb B}
\def\A{\Bbb A}
\def\I{\Bbb I}
\def\J{\Bbb J}
\def\N{\Bbb N}
\def\F{\Bbb F}
\def\l{\lambda}
\def\L{{\cal L}}
\def\c{\overline{\C}}
\def\a{\alpha}
\def\b{\beta}
\def\al{\alpha}
\def\ga{\gamma}
\def\z{\zeta}
\def\e{\epsilon}
\def\g{\gamma}
\def\G{\Gamma}
\def\d{\delta}
\def\m{\mu}
\def\n{\nu}
\def\e{\epsilon}
\def\f{\phi}
\def\D{\partial}
\def\BD{\bar{\partial}}
\def\r{\rho}
\def\2h{\H ^2}
%\def\H^3{\H ^3}
\def\h^*{\H ^2 _*}
\def\q.c.{quasiconformal~}
\def\PSL{PSL(2, \C )}
\def\sll{PSL(2, \C )}
\def\F{\cal F}
\def\Rs{Riemann surface~}
\def\Bd{Beltrami differential}
\def\bd{boundary~}
\def\qd{quadratic differential}
\def\bl{\blacksquare}
\def\Sd{Schwarzian derivative~}
\def\nbd{neighborhood~}
\def\BE{\begin{equation}}
\def\EE{\end{equation}}


\begin{document}
\title{Introduction to Teichm\"uller Theory}
\author{Michael Kapovich}
\date{August 31, 2008}
\maketitle


\section{Introduction}

This set of notes is based on a course I taught at University of
Utah in 1992--1993. It contains basic material on Riemann
surfaces, Teichm\"uller spaces and Kleinian groups.

\tableofcontents

\section{Riemannian and conformal geometry on surfaces.}

Conformal maps are smooth maps  in domains in $\c$ with derivatives
 in
$$CSO(2) = \R _+ \times SO(2)$$
The special feature of the complex dimension 1 is that  the classes of
 (locally) biholomorphic and (locally) conformal maps in subdomains of
$\bar \C$ coincide.


\begin{defn}
Riemann surface is a (connected)
1-dimensional complex manifold.
\end{defn}

Classes of (locally) biholomorphic and (locally) conformal maps in
$\c$ coincide. Therefore, each complex curve 1-1 corresponds
to a conformal structure on the 2-dimensional surface $S$
(maximal atlas with conformal transition maps).

Riemann surface with punctures $X$ is obtained from a Riemann surface  $\bar X$
by removing some discrete set of points. We mainly will be interested in
Riemann surface  $X$ of ``finite type'' $(g, p)$ which have $g=$ genus
 of compact surface $\bar X$;
$p=$ number of punctures.


Infinitesimally, conformal structure is a reduction of the prin\-ci\-pal
$GL(2, R)$ (frame) bundle $F(S)$ over $S$ to a principal bundle with the structure group $CSO(2)$. The Riemannian structure on $S$ is a reduction of $F(S)$ to a principal subbundle with the structure group $SO(2)$. Thus, each conformal structure can be obtained from
a Riemannian metric (this is true in arbitrary dimension). This is a general fact of the reduction theory: the quotient $CSO(n)/SO(n)\cong \R_+$ is contractible. Therefore we can use:

\begin{thm}
Suppose that $G$ is a Lie group and $H$ is its Lie subgroup so that $G/H$ is contractible.  Then for each manifold $M$ any principal $G$-bundle can be reduced to a principal $H$-subbundle.
\end{thm}

As we shall see, for the surfaces the converse is true as well:

\begin{thm}
 (Gauss' theorem on isothermal coordinates).
 For each Riemannian surface $(S, ds^2 )$ there exists
a local system of coordinates such that $ds^2 = \r (z) |dz|^2$.
 I.e. any Riemannian
metric in dimension 2 is locally conformally-Euclidean.
\end{thm}

Notice that  the system of coordinates on $S$ where $ds^2$ has the type
 $ds^2 = \r (z) |dz|^2$ is a
conformal structure on $S$. Really, the transition maps are
isometries between metrics $\l _1 |dz| , \l _2 |dz|$ on domains in
$\c$, thus they are conformal maps with respect to the
 Euclidean metric. Two metrics define one and the same
conformal structure if they are ``proportional''.

\begin{thm}
 (Uniformization theorem). For any Riemann surface $S$
 the universal cover
of $S$ is conformally- equivalent either to the (a) unit disc $\Delta$ or to
(b) $\C$ or to (c) $\c$.
\end{thm}

These classes of \Rs correspond to the following types:

(a) $(0, 0)$ (rational type),

(b) $(1, 0) , (0, 1), (0, 2)$ (elliptic type: torus, complex plane,
$\C ^* = \C - \{0\}$),

(c) other (hyperbolic type).


The proof of these theorems will be given later as a corollary
from some existence theorem in PDEs.

The groups of conformal automorphisms of $\Delta$, $\C$, $\bar {\C}$
consist of linear-fractional transformations. In the case (a)
our surface is simply connected.

The fundamental group $\Gamma$ of $X$ acts properly on $\tilde X$.
 Thus, in the case (b)
the group $\Gamma$ consists only of Euclidean isometries. As we shall
see later in the case (c) all conformal automorphisms preserve
the hyperbolic metric.

1) Torus. Metric can be obtained by identification of sides of Euclidean
rectangle.

2) $exp(\C ) = \C ^*$ - the universal covering.

We will be mainly interested in surfaces of ``hyperbolic type''.

Our strategy in proving U.T.: (1) use some geometry to construct
a complete hyperbolic metric on $S$. Then (2) use some analytic technique to
prove that each metric is conformally hyperbolic.

Hyperbolic plane: $\H ^2 = \{ z : Im(z) > 0 \}$ with the hyperbolic metric
$ds = |dz|/Im(z)$ (that has curvature $-1$). Recall that the group of biholomorphic automorphisms of the upper half-plane consists of linear-fractional transformations, i.e. equals $PSL(2,\R)$. Suppose that $f \in PSL(2, \R )$; then
$Im(fz) = Im(z) |f'(z)|$; thus, $f$ is an isometry of $\H ^2$.

\begin{defn}
A hyperbolic surface $X$ is a complete connected 2-dimensional Riemannian surface of the constant curvature $-1$.
\end{defn}

The universal cover $\tilde{X}$ of $X$ is again complete, hence it
is isometric to $\H^2$. Therefore we get an equivalent definition
of a hyperbolic surface:

\begin{defn}
Let $G$ be a properly discontinuous group of isometries of $\H^2$ which acts freely. Then $X= \H^2/G$  is a hyperbolic surface.
\end{defn}

We will use two models of the hyperbolic plane $\H^2$: the upper half-plane and the unit disk.
Geodesics in the hyperbolic plane are the arcs of Euclidean circles orthogonal to
$\D \H ^2$. Proof: use the inversion and the property that
between each 2 points the geodesic is unique.

{\bf Horoballs and hypercycles}. Horoballs in the unit disc model ($\Delta$) of the hyperbolic plane are Euclidean discs in $\Delta$ which are tangent to
the boundary of $\Delta$. If $h$ is a geodesic in $\H ^2$ then the boundary of its $r$-\nbd is called a
``hypercycle''.



\begin{defn}
({\bf Types of isometries}.) Consider a space $X$ of negatively
pinched sectional curvature $-b < K_X < a < 0$. Then an isometry
$g$ of $X$ is called {\bf elliptic} if it has a fixed point in
$X$. An isometry is called {\bf parabolic} if it has a single
fixed point in $\bar{X}= X\cup \D X$. An isometry is called {\bf
hyperbolic} (or {\bf loxodromic}) if it has exactly two fixed
points in $\bar{X}= X\cup \D X$.
\end{defn}

Examples: $z \to 2z$ (hyperbolic) ; $z \to z + a$   (parabolic);
Euclidean rotation of $\Delta$ around the center (elliptic).


\begin{rem}
Suppose that $p$ in a puncture on $X$. Then $p$ has a
\nbd $U$ which is conformally equivalent to a puncture on $\C /<\g >$
where $\g$ is a translation. Really, $U$ can be realized as a
\nbd of $0 \in \C$. Then the universal covering of $\C ^*$ is
$$
\exp : \C \to \C ^*$$
and the deck-transformation group is $< z \mapsto 2i\pi + z >$.
\end{rem}

{\bf Non-example.} Now suppose that the boundary of the strip
$$
1 \le  Re (z) \le 2 \ ; Im (z) > 0
$$
is identified by the homothety (hyperbolic transformation):
$$
h: z \mapsto 2z$$
denote the result by $A$.
Let's prove that the \nbd  $\D _1 A$ of $\infty$
  at $A$ isn't conformally equivalent
to a \nbd of $\infty$ in $\C$.

Notice that  $A$ is conformally isomorphic to
$\{ z \in \C : Re(z) > 0 , Im(z) > 0\} / <h>$.
Then $A \subset T^2 = \C ^* / <h>$ ; $\D _1 A$ is a curve on $T$.
There is a nondegenerate holomorphic map $f : T \to  \bar{\C}$;
$f(\D _1 A )$ is a smooth compact curve in $\C$. Suppose that
$q : U \to T$ is a biholomorphic embedding where $U$ is a closed
neighborhood of $\infty$ in $\C$;
$q(U)$ is a one-sided \nbd $N$ of $\D _1 A$.
Then
$f(N)$ is relatively compact in $\C$. Thus, the function
$q \circ f$ is bounded in $U$; then it extends holomorphically to
$\infty$. Thus, $qf(U \cup \infty )$ is compact and contains $q(\D _1 A)$.
However, it means that $qf(\infty ) \supset  q(\D _1 A)$ which is impossible.
{\bf QED of Non-example.}

\bigskip
Here is a way to construct a hyperbolic surface. Suppose that $P$ is a convex closed polygon
in $\H ^2$ and we have some isometric identifications of its sides
so that after gluing the total angle around each point is $2\pi$ and
the result of gluing is a surface (without boundary). Then $S= P/\sim$
has a natural hyperbolic structure. Unfortunately, this structure can
be incomplete.

\begin{ex}
 Put $A = \{ z \in \C : Im(z) > 0, 1< Re(z) \le 2 \}
\subset \H ^2$. Let $g: z \mapsto 2z$ ; then identify the sides of
$A$ by the equivalence relation: $z \cong 2z$. The surface $A /\cong$
is not complete.
\end{ex}

Let's try to figure out the criterion of the completeness. First assume
that $P$ is compact. Then $S$ is  complete. Now consider
closed finite-sided polygons $P$ of finite area.  Then, our problem
is reduced to the consideration of the isolated vertices which can be
the only source of incompleteness.
Let $\Gamma$ be a group generated by identifications.

\begin{thm}
(Criterion of completeness.)
 $S$ is complete iff the stabilizer of each vertex
 is parabolic.
\end{thm}

In particular we proved now the following. Under conditions above ($S$ is
complete)
the action of the group $\Gamma$ can be identified with the fundamental group
of $S$ and $P$ is the fundamental domain for $\Gamma$.

\begin{thm}
 Let $(l_1 , l_2 , l_3 ) \in \R  ^3 $ be nonnegative numbers. Then there exists a complete hyperbolic
structure $X$ with geodesic boundary on the pair of pants ($\S ^2
\setminus 3$~discs)  such that lengths of boundary curves are
$(l_1 , l_2 , l_3 )$.
\end{thm}
\proof Use 3 disjoint mutually nonseparating geodesics in $\H ^2$
such that hyperbolic distances between them are the numbers: $l_1
/2 , l_2 /2 , l_3 /2$ (the continuity principle). Then connect the
geodesics by orthogonal segments. This gives  a hexagon $Y$ in $\H
^2$ with right angles such that lengths of 3 sides are $(l_1 /2 ,
l_2 /2 , l_3 /2 )$.
 Take a double of $Y$ to obtain $X$. \qed

\begin{rem}
We allow some $l_j$ to be $0$, the corresponding boundary curves degenerate to punctures in this case.
\end{rem}

\begin{thm}
(Existence theorem for hyperbolic structures). Each surface of the
hyperbolic type has a complete hyperbolic structure.
\end{thm}

\proof Each surface with punctures can be split along disjoint
simple curves to a union of ``pairs of pants''. Find hyperbolic
structures on each component such that punctures correspond to
curves of $0$-length, other curves have one and the same length
(say $1$). Finally glue these pairs of pants together via
isometries of their boundary components. \qed

\section{Quasiconformal maps}

 The main analytic tool for proof of the Uniformization Theorem and
for all further discussion will be the theory of {\bf quasiconformal maps}.

\subsection{Smooth quasiconformal maps}

An orientation preserving homeomorphism $f$ of a domain
$A \subset \bar {\C}$ is $K$-quasiconfor\-mal iff the function
\BE
\label{1}
H(z) = \lim sup _{r \to 0} \frac {\max |f(z + r^{i\f}) - f(z)|}
{\min |f(z + r^{i\f}) - f(z)|}
\EE
is bounded in $A - \{\infty , f^{-1}\infty \}$ and $H(z) \le K$
a.e. in $A$.

Suppose in addition that $u_x v_y - u_y v_x =
|f_z |^2 - |f_{\bar z}|^2 = J_f(z) > 0$.
Denote
\BE
\label{2}
\D _{\a} f(z) = \lim _{r \to 0} \frac{f(z + re^{i\a} ) - f(z)}{re^{i\a}}
\EE
Then $\D _{\a} f(z) =  \D f + \BD f e^{-2i\a}$,
so
\BE
\label{3}
\max _{\a} |\D _{\a} f(z)| = |\D f (z)| + |\BD f (z)|
\EE
\BE
\label{4}
\min _{\a} |\D _{\a} f(z)| = |\D f (z)| - |\BD f (z)|
\EE

Recall that a function $\phi: \R \to \R^m$ is called {\em absolutely continuous}
if it has measurable derivative $\phi'$ almost everywhere (in the domain
$D$ of $\phi$) and for each subinterval $[a, b]\in D$ we have:
$$
\phi(b)- \phi(a) = \int_a^b \phi'(x)dx
$$

A function $\phi: D\subset \R^2 \to \R^2$ (defined on an open subset $D$) is called ACL (absolutely continuous on lines) if for almost every line $L$ the restriction $\phi|L\cal D$ is absolutely continuous.

\begin{rem}
In particular each ACL function has measurable partial derivatives a.e. in $D$.
\end{rem}

Suppose that $f$ is ACL  and (or has generalized $L^2$ partial derivatives).
Then, the $K$-quasiconformality is equivalent to the fact  that the quotient
\BE
\label{5}
D_f = \frac{\max _{\a} |\D _{\a} f(z)|}{\min _{\a} |\D _{\a} f(z)|} =
\frac{|\D f (z)| + |\BD f (z)|}{ |\D f (z)| - |\BD f (z)|}
\EE
is finite and a.e. bounded by $K$.

This is the same as:
$$
|\BD f(z) | \le \frac{K-1}{K+1}|\D f (z)|$$
Suppose in addition that $J_f(z) > 0$. Then we can form the {\bf complex dilatation of} $f$ :
\BE
\label{6}
\m (z) = \frac{\BD f(z) }{\D f(z)}
\EE
\BE
\label{7}
|\m (z)|  \le \frac{K-1}{K+1} < 1
\EE

The differential equation
\BE
\label{8}
{\BD f(z) } = \m (z) \D f(z)
\EE
is called the {\em Beltrami equation}. If $\m (z) = 0$ then it becomes the
Cauchy-Riemann equation. Each solution of the latter equation is holomorphic.

\medskip
{\bf Complex dilatation under the composition.} Let $\zeta = g(z)$, then
\BE
\label{9}
\m _{f\circ g^{-1}} (\zeta ) = \frac{\m _{f} (z) - \m _g (z)}
{1- \m _{f} (z) \overline{\m _g (z)}} (\frac{\D g(z)}{|\D g(z)|})^2
\EE

\begin{thm}
({\bf Existence-Uniqueness Theorem}.)
If $f, g$ are quasiconformal in $A$ with the same complex dilatation a.e.
then $f \circ g^{-1}$ is conformal.

For every measurable function $\m$ in the domain $A$ with $\|\m
\|_{\infty} < 1$ there exists a quasiconformal homeomorphism with
the complex dilatation $\m$.
\end{thm}

\begin{defn} A map  $f$ in $D$ is K- \q.c. if
\begin{itemize}
\item $f$ is a homeomorphism ;
\item $f$ is AC (absolutely continuous) on a.e. coordinate line in $D$ (ACL
property);
\item $|f_{\bar z}| \le k|f_z |$  where $k = \frac{K-1}{K+1} < 1$.
\end{itemize}
\end{defn}

\subsection{Properties of \q.c. maps}

1) Gehring- Lehto:  quasiconformal maps  $f$ are differentiable
 a.e. in $D$.

2) Partial derivatives are locally in $L^2$. And vice- versa:
if the partial derivatives are locally in $L^2$ then $f$ is ACL.

3) Area is absolutely continuous function under q.c. maps. Thus, $f_z \ne 0$
a.e. in $D$.

4) {\bf Mori's inequality}:

Let $\Omega \subset \bar{\C}$ , $f: \Omega \to \Omega '$. Normalize $f$
so that $f(\infty ) = \infty$, $f(a_j ) = b_j$ , $j= 1, 2$.

Then for every compact subset $G \subset \Omega \cap \C$ we have
the Holder inequality:
\BE
\label{10}
|f(z) - f(w)| \le M_G |z - w|^{1/K}
\EE

5) {\bf Convergence property}:
 Suppose that $f_n$ be a sequence of
$K_n$-q.c. mappings of $\c$ so that :

(a) $f_n$ fix three points: $\infty$ , $a_1 \ne a_2 \in \C$;

(b) $K_n \le K < \infty$.

Then $f_n$ has a subsequence which is uniformly convergent on compacts to a \q.c. homeomorphism.

6) {\bf Extension property.} Suppose that $f : \2h \to \2h$ is a \q.c.
self-map of the upper half-plane. Then $f$ extends
to $\c$ to  a \q.c. homeomorphism, whose restriction
to  the line $\D \2h$ is quasisymmetric if we normalize it by
$f(\infty ) = \infty$ :
\BE
\label{11}
C^{-1}  \le \frac{\phi (x + t ) - \phi (x)}{\phi (x  ) - \phi (x- t)}
\le C
\EE
However, the boundary value isn't necessarily AC. According to
Ahlfors and Beurling
the condition (11) is also
sufficient for the extension of $\phi$ to a \q.c. map of $\c$.

\medskip
See proofs in \cite{Ahlfors}.


\subsection{The existence theorem}
Consider the Beltrami equation: \BE \label{12} f_{\bar z} = \m f_z
\EE where $\mu \in L_\infty$ and $\|\m\| \le k < 1$.

Recall that for $f$ with $L^1$-derivatives we have:
\BE
\label{13}
f(\zeta ) = -\frac{1}{\pi} (P.V.)\int_{D} \frac{f_{\bar z}}{z- \zeta} dx dy +
\frac{1}{2i\pi}\int_{\D D} \frac{f(z)}{z- \zeta}dz
\EE
(generalized Cauchy formula). Here P.V. means the principal value
in the sense of Cauchy. In particular,
\BE
\label{14}
\bar{\zeta} = -\frac{1}{\pi} (P.V.)\int_{\Delta _R} \frac{1}{z- \zeta} dx dy
\EE
Consider the operator $P$ on the functions $h \in L^p$, $p > 2$,
\BE
\label{15}
Ph(\zeta ) = -\frac{1}{\pi} (P.V.)\int_{\C} h(z) \frac{\zeta}{z(z- \zeta)}
dx dy
= -\frac{1}{\pi} \int_{\C} h(z) (\frac{1}{z} - \frac{1}{z- \zeta}) dx dy
\EE
The integral is correctly defined as (P.V.) since $h \in L_p$ in a compact
domain implies that $f \in L_1$.

\begin{lem}
$Ph$ is continuous and satisfies the uniform Holder
inequality with the exponent $(p-2)/(p-1)$.
\end{lem}
\proof $h \in L_p$, $\zeta (z(z- \zeta))^{-1} \in L_q$ where $1/p
+ 1/q = 1$, $1 < q  < 2$. Then the Holder inequality implies that:
\BE \label{16} |Ph(\zeta )| \le \frac{1}{\pi} \|h\|_p
\|\frac{|\zeta |}{z(z- \zeta )}\|_q \EE \BE \label{17} \int
\frac{|\zeta |^q}{|z(z- \zeta )|^{q}} dx dy = |\zeta |^{2 - q}
 \int_{\C} |z(z - 1 )|^{-q}dx dy =  |\zeta |^{2 - q} K_p
\EE Then: \BE \label{18} |Ph(\zeta )| \le |\zeta |^{(p- 2)/(p-1)}
K_p \|h\|_p \EE and if $h_1 (z) = h(z + \zeta _1 )$ then: \BE
\label{19} Ph_1 (\zeta _2 - \zeta _1 ) = Ph(\zeta _2 ) - Ph(\zeta
_1 ) \EE so $Ph$  is Holder with the exponent  $(p-2)/(p-1)$. \qed

\begin{rem}
 The formula \ref{16} implies that $Ph(0) = 0$.
\end{rem}

\bigskip
The operator $T$ is defined for $h \in C^2 _0 $:
\BE
\label{20}
Th(\zeta ) = \lim_{\e \to 0} - \frac{1}{\pi} \int _{|z- \zeta | > \e }
\frac{h(z)}{(z- \zeta)^2}dx dy
\EE

This operator is called ``Hilbert transformation''.
 Notice that $Th(\zeta ) = O( |\zeta |^{-2})$ as $\zeta \to \infty$, since
\BE
\label{21}
|Th(\zeta )| \le (\int_{D_R} h )\cdot \sup_{D_R} \frac{1}{|z- \zeta |^2} =
(\int_{D_R} h )\cdot {|\zeta - R |^2} = O(|\zeta |^{-2} )
\EE


\begin{lem} For $h \in C_0 ^2$ , $Th$ has class $C^1$ and:
\BE \label{22} (Ph)_{\bar{z}} = h  \ ; \hbox{i.e.\ \ } \ \ \D
\circ P = id \EE \BE \label{23} (Ph)_{z} = Th \ \ \ ;
(Th)_{\bar{z}} = (Ph_z )_{\bar{z}} = h_z \EE \BE \label{24} \int
|Th|^2 dx dy = \int |h|^2 dx dy \EE and moreover, $\|Th\|_p \le
C_p \|h\|_p$ for any $p > 1$ so that \BE \label{25} \lim_{p \to 2}
C_p = 1 \EE (Calderon- Zygmund inequality). Thus we can extend $T$
to $L_p$.
\end{lem}
\proof
 We shall skip the proof of the Calderon- Zygmund inequality,
the reader can find it in \cite{Ahlfors}.

(i) The generalized Cauchy formula (\ref{13}) implies that
$$
(Ph)_{\bar z} = - \frac{1}{\pi} \int_{\C} \frac{h_{\bar z}}{z- \zeta} dx dy
$$
\BE
\label{26}
(Ph)_{\bar{ z}} = - \frac{1}{\pi} \int_{\C} \frac{h_{z}}{z- \zeta} dx dy
\EE
Thus,
$$
- \frac{1}{\pi} \int_{\C} \frac{h_{\bar z}}{z- \zeta} dx dy =
- \frac{1}{2\pi i} \int_{\C} \frac{dh dz}{ z- \zeta} $$
$$
= \lim_{\e \to 0} \frac{1}{2\pi i} \int_{|z - \zeta| = \e}
\frac{hdz}{ z- \zeta}
= h(\zeta)$$

(ii)
\BE
\label{27}
P(h_{z}) = \frac{1}{2\pi i}\int_{\C} \frac{dh d\bar{z}}{z- \zeta} =
\EE
\BE
\label{28}
\lim _{\e \to 0} ( -\frac{1}{2\pi i}\int_{|z - \zeta| = \e}
 \frac{h d\bar{z}}{z- \zeta} )
+ \frac{1}{2\pi i}\int_{\C} \frac{h dz d\bar{z}}{(z- \zeta )^2}
 = Th(\zeta )
\EE

\begin{rem}
\label{2.5}
 It follows that $Ph$ is holomorphic near $\infty$
and $\cong a_0 + \frac{a_1}{z}$ as $z \to \infty$ since $\D Ph = Th
\cong z^{-2}$. Thus, if $a_0 = 0$, then
$$Ph \in L^p (\{z \in \C : |z| \ge R \})$$
\end{rem}

\bigskip
(iii) Now, let's prove that $T$ is $L_2$-isometry.
$$
\int_{\C} |Th|^2 dz d\bar{z}  =
\int_{\C} (Th) (\overline{Ph})_{\bar{z}} dz d\bar{z} $$
$$
=  \int (Th \overline{Ph})_{\bar{z}} dz d\bar{z}
 - \int (Th )_{\bar{z}}\overline{Ph} dz d\bar{z}
=
-\frac{1}{2i} \int h_z \overline{Ph}  dz d\bar{z} $$
because $(Th)_{\bar{z}} = h_z$ and
\BE
\label{29}
\int (Th \overline{Ph})_{\bar{z}} dz d\bar{z} =
\int_{\D D_R} Th \overline{Ph}dz \to 0
\EE
since $Th = O(|z|^{-2})$ as $|z| \to \infty$.

On another hand, \BE \label{30} \int_{\C} h \bar{h} dz d\bar{z} =
\int_{\C} h (\overline{Ph})_z = \int_{\C} (h \overline{Ph})_z -
\int_{\C} h_z \overline{Ph} \EE the 1-st term is approximately
equal to \BE \label{31} \int_{\D D_R} h \overline{Ph} d\bar{z} = 0
\EE since $h$ has a compact support. So, $T$ is an isometry. \qed


\begin{thm}
 If  $\mu$ has a compact support then there exists
a unique solution $f$ of the Beltrami equation
 such that $f(0) = 0$ and $f_z - 1
\in L_p$ where $p$ is such that $C_p \|\mu \|_{\infty} \equiv C_p
k < 1$. Such solution $f = f_{normal}^{\m }$ is called ``normal''.
\end{thm}
\proof
(a) {\bf Uniqueness}. Suppose that $f$ is a solution. Then $f_{\bar z} \in L_p$
(because it has a compact support and locally it's roughly
proportional to $f_z$)
and there exists $P( f_{\bar z} )$ so $P( f_{\bar z} )(0) = 0$.
 Then the function
\BE
\label{32}
F = f - P(f_{\bar z} )
\EE
is analytic.
 Then $f_z - 1 \in L^p$  implies  that $F_z - 1 = f_z - 1 - T (f_{\bar z} ) $
is in $L^p$ since the last term has quadratic decay at infinity. Hence,
$F_z = 1$ and $F = z$ since $F(0) = 0$.

 Thus $f = P(f_{\bar z} ) + z$ and $f_z = T(\mu f_z ) + 1$. Let $g$ be another
solution. Then $f_z - g_z = T(\mu (f_z - g_z ))$, hence \BE
\label{33} \|f_z - g_z \|_p \le kC_p \|f_z - g_z \|_p \EE and $f_z
= g_z$ , $f_{\bar{z}} = g_{\bar{z}}$ so $f = g$.

(b) {\bf Existence}.  Consider the equation: \BE \label{34} h =
T(\mu h) + T\mu \EE The linear operator $h \mapsto T(\mu h)$ on
$L_p$ has norm $\le kC_p < 1$. Then the series \BE \label{35} h =
T\mu + T\mu (T\mu ) + T\mu (T\mu (T\mu )) ... \EE is convergent in
$L_p$. This is a solution of (\ref{34}). Then, for this $h$ the
function \BE \label{36} f = P[\mu (h +1)] + z \EE is the solution
since  $\mu (h + 1)$ is in $L_p$; \BE \label{37} f_{\bar{z}} = \mu
(h+1) \ \ \ ; f_z = T[\mu (h +1)] + 1 = h +1 \EE and $f(0) = 0$
and $f_z - 1 = h \in L_p$. \qed


\begin{rem}
 It follows from the formulas (\ref{37}) that $\D f = T(\BD f ) + 1
= T(\m \D f ) + 1$.
\end{rem}

\begin{lem}
 If $\nu _n \to \mu$ uniformly a.e.
 and supports are bounded; then
\BE \label{38} \|\D g_{normal}^{ \nu _n}  - \D f_{normal}^{ \mu}
\|_p \to 0 \EE and $g_{normal}^{\nu _n} \to f_{normal}^{ \mu}$
uniformly on compacts. Moreover, since these functions are
holomorphic near $\infty$, the convergence is uniform on $\c$.
\end{lem}
\proof Put $g_n := g_{normal}^{ \nu _n}$. The Remark above implies that
\BE
\label{39}
\D f - \D g_n = T(\mu \D f_z - \D \nu g_n )
\EE
and hence
$$
\|\D f - \D g_n \|_p \le \|T(\nu _n ( \D f - \D g_n ))\|_p +$$ \BE
\label{40} \|T(\mu - \nu _n )\D f \|_p \le kC_p \|\D f - \D g_n
\|_p + C_p \|(\mu - \nu _n )f_z \|_p \EE

Thus, $\|\D f - \D g_n \|_p (1 - k C_p ) \le C_p \|(\mu - \nu _n
)f_z \|_p \to 0$.

This implies the statement about convergence of derivatives (since
$f_z \in L^p _{loc}$). The Beltrami equation implies the $L^p$ convergence of
the $\bar{\D}$ -derivatives.

Now consider $f = P(\mu h_\mu + 1) + z = P(f_{\bar{z}}) + z$ Then
$|f - g| = |P( f_{\bar{z}} - g_{\bar{z}})| \le K_p \|f_{\bar{z}} -
g_{\bar{z}}\|_p |z|^{2 - q}$ which implies the last assertion.
\qed

\begin{lem}
If $\m$ has a compact support and distributional derivative
 $\mu _z \in L_p$ ($ p > 2$)
 then the normal solution
$f \in C^1$ and is a homeomorphism.
\end{lem}
\proof Let's try to determine $\lambda$ such that the system:
\BE
\label{41}
f_z =  \lambda \ \ \ ; f_{\bar {z}} =  \lambda \mu
\EE
has a solution (which will be the solution of the Beltrami equation).
The necessary and sufficient condition is that
\BE
\label{42}
\lambda _{\bar{z}} = (\lambda \mu )_z = \lambda _z \mu + \lambda  \mu _z
\EE
Or, for $\sigma := log \lambda$:
\BE
\label{43}
(\sigma  ) _{\bar{z}} = \mu ( \sigma )_z + \mu _z
\EE
Consider the operator:
$$
T_{\mu} : h \mapsto T(\mu h ) \eqno(44)$$
The $L^p$ norm of it is less than 1, hence in $L^p$ we have:
$$
(T_{\mu} - 1 )^{-1} = 1 + T_{\mu} +T_{\mu} ^2 + ... \eqno(45)$$
Therefore, we can find $q \in L_p$ such that
$$
q = T(\mu q) + T(\mu _z ) \eqno(46)$$
Put $\sigma =  P(\mu q + \mu _z ) + const$
 so that $\sigma \to 0$ as $z \to \infty$.
Thus, $\sigma$ is Holder continuous and
$$
\sigma _{\bar{z}} = \mu q + \mu _z \ \ ; \sigma _z = T(\mu q + \mu _z ) = q
\eqno(47)
 $$
Hence $\lambda = \exp (\sigma)$ satisfies the equation and there is a solution
$f$ of the class $C^1$ and  we can normalize
$f(0) = 0$ so that
$f_z = \l \to 1$ as $z \to \infty$. Then $\l - 1 \cong \sigma (z) =
P (\mu q + \mu _z ) + const$ as $z \to \infty$. Thus, we can use the Remark \ref{2.5} to show that $\l - 1 \in L^p$ and $f$ is a normal solution.

The Jacobian
$$
|f_z |^2 - |f_{\bar z}|^2 = (1- |\mu |^2 )exp(2\sigma )\eqno(48)$$
is positive, hence, $f$ is a diffeomorphism. \qed

\begin{cor}
 For any $\mu$ with compact support the normal solution
is a homeomorphism.
\end{cor}
\proof We can approximate any $\mu \in L_{\infty}$ by smooth $\m
_n$ with compact support, solutions $f_{normal}^{ \m _n }$ are
diffeomorphisms and then $f_{normal}^{ \m _n }$ are convergent to
$f_{normal}^{ \m}$ uniformly on compacts. Therefore, we can use
the property (5) of q.c. maps (compactness property) to prove that
the limit is a homeomorphism. \qed

\bigskip

Now we need a  formula for composition of $f$ with Moebius maps $g$.

Namely, (under assumption $\zeta = h(z)$) it follows from
$$
\m _{f\circ h^{-1}} (\zeta ) = \frac{\m _{f} (z) - \m _h (z)}
{1- \m _{f} (z) \overline{\m _h (z)}} (\frac{\D h(z)}{|\D h(z)|})^2
\eqno(49)
$$
 that:
$$
\m _{g^{-1} f g} (\zeta ) = \m _f (g(\zeta )) \frac{\overline{\D g(\zeta )}}
{\D g(\zeta )} =: g_* \m \eqno(50)$$

\bigskip
{\bf Convention.} Now, by $f^{\mu}$ we shall
 mean the solution which fixes $0, 1, \infty$. It is called the ``normalized''
solution.

\bigskip
\begin{thm}
 For any $\m$ on $\C$ with the norm $\|\m \|_{\infty} < 1$
there exists a normalized solution of the equation $\BD f = \m \D f$.
This solution is a homeomorphism.
\end{thm}
\proof (a) The solution exists if $\m$ has a compact support, just adjust
the normal solution by Moebius transformation.

(b) Suppose that $\m$ vanishes in $\Delta _r (0)$. Then take $\nu =
g_* \mu$ where $g(z) = z^{-1}$. Then $\nu$ has a compact support and there
exists $h = f^\nu$. Now take $f = g h g^{-1}$ and $\mu _f = g^{-1} _* \nu
= \mu$.

(c) Consider the general case. We can decompose $\m$ as $\mu _1 + \mu _2$
where $\mu _1$ has a compact support $\Delta$ and $\mu _2$ has support
outside $\Delta$. Try to find the solution
$$
f^{\m} = f^{\l} \circ f^{\m _2} \eqno(51)$$
i.e. $g := f^{\m _2}$,
$$
f^{\l} = f^{\m} \circ g ^{-1} \eqno(52)$$
Then, according to (9) we have:
$$
\l = [(\frac{\mu - \mu _2 }{1 - \mu \overline{\mu _2}})
 (\frac{\D g}{|\D g|} )^2 ] \circ g^{-1} \eqno(53)
$$
Then, since $\m -  \mu _2 = \mu _1$ has compact support and, by convention,
$g$ fixes some \nbd ~ of $\infty$, the characteristic $\l$ also has a
compact support. So, there exists $f^\l$.
Put $ f= f^{\l} \circ f^{\m _2}$; $\nu = \mu _f$. We have to show that
$\nu = \mu$,
 and we can find $f^\m$ from (51). Notice that $\mu$ and
$\nu$ satisfy to one and the same equation (53). Suppose that
$\mu \ne \nu$ on some set $E$ of nonzero measure. The equation:
$$
\frac{\mu - \mu _2 }{1 - \mu \overline{\mu _2}} = \phi$$ is linear
by $\m$, thus the nonuniqueness implies that on $E$ we have: $\phi
\equiv - \m _2  $ and $1 \equiv - \phi \overline{\mu _2 }$; i.e.
$|\mu _2 |^2 \equiv 1$ on the set $E$ which is impossible since
$\|\mu _2 \|_{\infty} < 1$. Thus, $\mu = \nu$ and $f = f^\m$. \qed

\subsection{Analytical dependence of $f^\m$ on the complex dilatation}

\begin{thm}
 The normal and normalized solutions of the Beltrami
equations depend holomorphically on $\mu$ which means that
the map
$$
Belt: \mu \in B(1) \subset L_{\infty} \mapsto f^{\mu} \in
C^0(\Delta , \C )$$ has complex derivative for any disc $\Delta
\subset \C $; where $C^0(\Delta , \C )$ is the space of continuous
$\C$-valued functions with supremum norm; $B(1) \subset
L_{\infty}$ is the open unit ball.
\end{thm}
For proof see \cite{Ahlfors}. We shall need and prove
much weaker statement:

Denote by $\Delta (r) $ the open disc of radius $r$ in $\C$ with center at $0$.

\begin{thm}
  Let $\mu , \nu \in B(1)$ have compact support. Then
for  each $z \in \c$, the function
$$
w \in  \Delta (\e =  \frac{1 - \|\nu \|_{\infty}}{\|\mu
\|_{\infty}})
 \mapsto f_{normal}^{ w\m + \nu }(z)$$
is holomorphic.
\end{thm}
\proof
Recall the representation for the normal solution (36 -37):
$$
f(z) = P[(w\mu + \nu ) ( 1 +  T(w\mu  + \nu ) +
T[(w\mu + \nu )(T( w\mu  + \nu )] +  ... )]
+ z =$$
$$
 P[(w\mu + \nu ) ( 1 + A_1 (w) + A_2 (w ) + ... )]=
P[(w\mu ( 1 + A_1 (w) + A_2 (w ) + ... )    ]
 +$$
$$
 P[  \nu  ( 1 + A_1 (w) + A_2 (w ) + ... )] + z$$
 this series is uniformly convergent for all $w \in  \Delta (\e )$
and each term is a holomorphic (in fact, polynomial) function on
$w$. Then  each $z \in \c$, the limit depends holomorphically on
$w$. \qed

\begin{exe}
 Suppose that $D$ is a domain in $\c$ such that:
 each conformal automorphism of $D$ is Moebius. Then any \q.c. automorphism
of $D$ can be extended to $\c$.
\end{exe}

\def\la{\lambda}
\def\Om{\Omega}
\def\Ga{\Gamma}
\def\La{\Lambda}

\section{Quasiconformal  maps on Riemann surfaces}

 Let $X$ be a Riemannian surface, $ds^2$ is  a metric.
Then locally we can write
$$
ds^2 = Edx^2 + 2Fdxdy + Gdy^2$$
Put $dz = dx + idy$, $d\bar{z}
 = dx - idy$. Thus,
$ds= \la |dz + \m d\bar{z}|$,

where $\la ^2 = (E + G + 2\sqrt{EG - F^2})$,
$$
\m = \frac{E - G + 2iF}{E + G + 2\sqrt{EG - F^2}}$$
Notice that
$$
|\m |^2 =  \frac{E + G - 2\sqrt{EG - F^2}}{E + G + 2\sqrt{EG - F^2}} < 1$$

\begin{thm}
 (Gauss' theorem on isothermal coordinates).
 For each Riemannian surface $(S, ds^2 )$ there exists
a local system of coordinates such that $\mu = 0$. I.e. any Riemannian
metric in dimension 2 is conformally- Euclidean.
\end{thm}
\proof Suppose that $w= f^\m (z)$ is the q.c. homeomorphism with
the dilatation $\m$. Then $| dw | = |\D w dz + \BD w d\bar{z}| =
|\D w| |dz + \m d\bar{z}|$; thus $ds = \l |dw|/|\D w|$. \qed

\bigskip
Thus, we should look more carefully on the $\m _f$ as an object on
Riemann surface. The formula (50) shows that
 $\mu dz/{\overline {dz}}$ is a differential of the type $(-1, 1)$
on the surface $X$, i.e. $\m dz \otimes \D/\overline {\D z}$.  Such differential will be called a {\bf conformal
structure} since (a) each conformal class of Riemannian metrics
 on $S$ gives us some $\m$
and (b) given $\m$ we can solve the Beltrami equation and the maps $f^{\m}$
will define a complex (conformal) structure:

\medskip
For each point $z\in X$ we have a \nbd $U$ where we can solve the Beltrami equation $\bar\D f= \m \D f$, where $f: U\to \C$ is a homeomorphism.

If $f_i, f_j: U \to \C$ are solutions of the Beltrami equation then by uniqueness of solution $f_i \circ (f_j)^{-1}$ is holomorphic.

As the result we get a conformal structure on $X$ corresponding to
$\mu$. We will retain the notation $\mu$ for this conformal
structure. Automorphisms of $\mu$ are the self-diffeomorphisms $h$
of $X$ such that iff $h^* (\m ) = \m$. If  $\m$ is defined in some
domain $D$ in the plane $\bar \C$, then this condition means that:
$$
\m (z) \equiv \m (hz) \overline{h'(z)}/h'(z) \eqno(54)$$
where we assume $h \in Mob(\C )$.
Let $f_\m$ be the solution of the B.e.: where $\mu = 0$ outside $D$.
Then $f : (\bar {\C} , \m ) \to (\bar\C , can)$ is conformal.
Therefore, since $h \in Aut (\m )$, then $f\circ h \circ f^{-1}$ belongs
to $Mob (\C )$.

\begin{cor}
 Suppose that $G \subset \sll$;
$f_{\m}$ is a \q.c. map of $\c$
so that (54) holds for each $h \in G$. Then  $f_{\m} \circ h \circ
f_{\m} ^{-1} \in \sll$.
\end{cor}

\begin{cor}
 Suppose that $f: A\subset \c  \to f(A) \subset
\c$ is a \q.c. homeomorphism and $G$ is a subgroup of conformal automorphisms
of $A$ and (54) holds for each $h \in G$. Then    $f_{\m} \circ h \circ
f_{\m} ^{-1}$
is a conformal automorphism of $f(A)$ for each $h \in G$.
\end{cor}

\begin{thm}
\label{qc equivalence}
 Let $X, Y$ be surfaces of the same type. Then
there is a  \q.c. map $X \to Y$.
\end{thm}
\proof  Let $\bar{X}, \bar{Y}$ be conformal compactifications of
$X, Y$; then there is a diffeomorphism $f : \bar{X} \to \bar{Y}$
which maps punctures to punctures. The restriction of $f$ to $
\bar{X}$ is the desired \q.c. homeomorphism.  \qed

\bigskip
{\bf Formula for transformation of the Beltrami differential under} complex conjugation $g: z \to \bar{z}$.
If $f = f^\m$ and $h = g\circ f \circ g$ then
$$
\mu _h = g \circ \mu \circ g \eqno(55)$$
i.e. $\mu _h (z) = \overline{\mu (\bar{z})}$.


\section{Proof of the Uniformization Theorem for surfaces
of finite type}

 Let $X$ be a \Rs of finite type.
Then we can construct a hyperbolic surface $X_0$ of the same type.
denote by $f$ a q.c. homeomorphism $f: X_0 \to X$ (which exists due to Theorem \ref{qc equivalence}). Let $\mu$ be the dilatation of $f$ lifted in $\Delta$. Extend $\m$
to $ext (\Delta )$ by the symmetry. Put $\m = 0$ on $\D \Delta$.
Then the resulting differential Beltrami $\nu$ will be compatible with the
action of the group $\Gamma _0 = \pi _1 (X_0 )$. Therefore, $f^{\nu}$
conjugate $\Gamma _0$ to a group $\Gamma \subset PSL(2, \C )$ and since
$f^{\nu}$ commutes with conjugation, $\Gamma \subset PSL(2, \R )$.
However, the universal cover $\tilde X$ of the surface  $X$ is
biholomorphic to $(\Delta , \mu )$ which is conformally equivalent
to $\Delta$ via $f$. Thus, we proved the  Uniformization Theorem
 for surfaces of
hyperbolic type. The proof for the elliptic type is essentially
the same. Proof for the rational type is just a particular case of
the existence theorem for \q.c. maps. \qed

\section{Elementary theory of discrete groups.}
\subsection{Definitions}

Let $\S^k$ be  a round sphere in $\S ^{k+1} =
\overline{\R ^{k+1}}$. Then the inversion $J$ in $\S ^k$ defined  as follows.
If $\S ^k$ is an extended Euclidean plane, then $J$ is just the Euclidean
symmetry in $\S ^k$. Otherwise, if $O$ is the center of $\S ^k$,
 $x \in ext (\S ^k )$,
$L$ is the Euclidean line through $x, O$ and $K$ is the tangent cone from
$x $ to $\S ^k$,
then $J$  maps  $x$ to the orthogonal projection of $K \cap \S ^k$ to $L$.

It is easy to prove that each element $\gamma \in \sll$ is a composition of
even number of inversions.

Consider the group $G = \PSL$ acting on $\c$. Extend this action
in $\R ^3 _+ = \H ^3$ using inversions. Namely, if $\gamma \in
\sll$ is  a composition $J_1 \circ ... \circ J_s$ of inversions
in the circles $\sigma _j \subset \c$ , then each
$J_k$ is extended canonically to the inversion $\tilde{J}_k$
in the Euclidean sphere $\Sigma _j$ which contains $\sigma _j$ and
is orthogonal to $\c$. Then, define the extension $\tilde{\gamma}$ as
the product of extensions $\tilde{J}_1 \circ ... \circ \tilde{J}_s$.
It's easy to see that this extension doesn't depend on the decomposition
of  $\gamma$ in the product of inversions.

The extended complex plane $\c$ can be identified with $\S ^2$ via
stereographic projection. This defines on $\c$ the metric
of constant positive curvature $|dz|/(1 + |z|^2)$.
We can describe  $ \H ^3$ as $G/K$ where $K$ is the maximal
compact subgroup $SO(3)$. Three types of isometries of $\H^3$ can be distinguished by the matrices in  $SL(2,\C)$ representing these isometries:

\medskip
elliptic: $Tr(g) \in (-2 , 2)$;

\smallskip
parabolic: $Tr(g) = \pm 2$;

\smallskip
loxodromic: $Tr(g) \notin [-2, 2]$.

\medskip
A special type of loxodromic elements are {\it hyperbolic}
elements, which have real trace. They can be characterized
as elements with invariant Euclidean discs in $\c$.

The group $G$ has the
``Cartan decomposition'': $G = K A K$ where
$$
A= \{ a: z \mapsto k z , k \in \C^* \}$$

One can prove the existence of this decomposition geometrically. Namely,
$\H ^3 = X = G/K$ and $G$ acts on $X$ transitively on the right.
Let $x_0 \in X$ be the class of $K$; let $x \in X$ be any point. Then
there is an element $k \cdot a \in K A$ such that $ak (x) = x_0$
(if $\gamma$ is the invariant geodesic for $A$ then there is an element
$k \in K$ such that $k(x) \in \gamma$, then the action of $A$ translates
$k(x)$ into $x_0$). Therefore, for each $g \in G$ we can find $k, a$
such that: $ak g K = K$, since $gK = x, K = x_0$; thus $g \in KAK$.

\subsection{The convergence property}

{\em A quasiconstant map} $z_x : \c \to \c$ is a map such that for
$z, x \in \c$ we have:
$$
 z_x (w) = z \hbox{~~for each~~} w \ne x
$$
We let $z_x^{-1}:= x_z$. Each quasiconstant map naturally extends to
$\H ^3$. Let  $\hat{G} := G \cup \{ quasiconstants\}$

{\bf Topology.} A sequence of elements $g_n\in \hat{G}$ is convergent to
a quasiconstant $z_x$ iff $g_n$ converges to $z_x$ uniformly
on compacts in $\c - \{ x \}$.

\begin{exe}
$g_n \to g$ iff $g_n ^{-1} \to g^{-1}$.
\end{exe}

\begin{thm}
 $\hat{G} = G \cup \{ quasiconstants\}$ is compact.
\end{thm}
\proof  For any sequence $g_n$ we have $g_n = k_n a_n c_n$ where
$k_n , c_n \in K$, $a_n \in A$. Up to subsequence we can assume
that $k_n \to k$, $c_n \to c$, $a_n \to a$ where $a$ is either
element of $A$ or quasiconstant $\infty _0$. Then $g_n$ is
convergent on $\c - c^{-1} (0)$ to $k (\infty )$. Thus, $g_n \to
a_b$ where $a =  k (\infty ) , b= c^{-1} (0)$. \qed

\subsection{Discontinuous groups}

A subgroup $\G$ of $\sll$ is called {\it elementary} if
it has either an invariant point in $\H^3 \cup \c$ or invariant geodesic
$L$ in $\H^3$ (in the latter case $\G$ can change the orientation on $L$).

Examples of elementary groups: (i) Consider the group $B\subset SL(2,\C)$
which consists of  upper-triangular matrices. Let $PB$ is the projection of $B$
to $PSL(2,\C)$. Then each subgroup of $PB$ is elementary (since $PB$ fixes the
point $\infty$ of $\bar\C$.

(ii) Let $\Ga$ be a finite subgroup of $Isom(\H^n)$. Then $\Ga$ has a fixed point in $\H^n$ (hint: consider the $\Ga$-orbit $\Ga p$ of a point $p\in \H^n$,
take the smallest metric ball $D$ in $\H^n$ which contains $\Ga p$; then the center of $D$ is $\Ga$-invariant).

\begin{defn}
 Let $\G \subset G$. Then $x \in \c$ is a
{\em point of discontinuity} for $\G$
 if there is a \nbd \ $U$ of $x$ such that $U \cap gU \ne \emptyset$ only for
finitely many $g \in \G$. Usually, this means that $U \cap gU \ne \emptyset$
for all $G - \{ 1\}$; the exceptional case is: $x$ is a fixed point of a finite
subgroup $F \subset \G$.
\end{defn}

The {\em domain of discontinuity} $\Omega (\G )$ consists of all
points of discontinuity. Clearly this is an open subset of $\c$. If $g$ is an element of $\Ga$ which has infinite order then the fixed-point set of $g$ is disjoint from $\Om(\Ga)$.

\begin{defn}
A {\bf discontinuous} (another name is  {\bf Kleinian}) group is a
subgroup of $\sll$ with nonempty discontinuity domain.
\end{defn}

More general concept is a {\em discrete group}, i.e. a subgroup
$\G$ of $\sll$ which is a discrete subset in the induced topology (i.e. if
$\G \ni \gamma _n \to \gamma \in \sll$ then $\gamma _n = \gamma$ for
all but finite elements of the sequence $\{ \gamma _n \}$.

\begin{exe}
$\G$ is discrete iff $1$ is an isolated point of $\G$. (Hint: if
$\G$ isn't discrete, consider the sequence $\gamma _{n+1} \gamma _n ^{-1}$).
\end{exe}

Clearly all Kleinian groups  are discrete. Therefore all their
elliptic elements have finite orders (however there is no a priori
bound on these orders). One can show that if $\G \subset PSL(2, \R
)$ is nonelementary then the absence of elliptic elements of
infinite order is also a {\em sufficient} condition for
discreteness. However, the discreteness doesn't imply that the
group is Kleinian.

\begin{exe}
Consider $\Ga:= PSL(2, \Z [i])$ where $\Z [i]$ is the ring of ``Gaussian
integers'' (i.e. complex numbers of the form $x + iy$ where $x, y\in \Z$).
Show that $\Ga$ is discrete. Prove that the domain of discontinuity  of $\Ga$
is empty. (Hint: show that each rational Gaussian number is a fixed point
of a parabolic element of $\Ga$.)
\end{exe}

\begin{thm}
(Schur's Lemma.) Suppose that $\Ga$ is a finitely generated {\bf
torsion} subgroup of $GL(n,\C)$ (i.e. each element of $\Ga$ has
finite order). then $\Ga$ is finite.
\end{thm}

This lemma immediately implies that if a discrete group $\Ga$ consists only of elliptic elements, then $\Ga$ is finite (and hence elementary).

However, there are examples of infinite (nondiscrete) subgroups
$T \subset Isom(\H^5)$ such that each element
$t \in T$ has a fixed point in $\H ^5$ but the group $T$ doesn't have a
fixed point  in $\H ^5$. Nevertheless, such group necessarily has a fixed
point in $\H ^5 \cup \S ^4$. To construct such example, take a free
2-generated subgroup $<a , b> = H \subset SU(2) \subset SO(4)$ and
$v, w \in \R ^4 - \{0 \}$; $g (x) = ax + v , f(x) = bx + w$ for $x \in \R ^4$.
Then each element of the free group $T = <g, f > \subset Isom (\E ^4 )$ has
a fixed point in $\R ^4$ but there is no global fixed point for the action
of $T$ in $\E ^4$. The extension of $T$ in $\H ^5$ provides the desired
example. Such examples are impossible for $\H^k$, $k < 5$.

\bigskip
Now we can define the {\it limit set} $\Lambda (\G )$
of a Kleinian group $\G$ as the
set of accumulation points for the orbit  $\G x $ for some
 $x \in \Omega (\G )$ (i.e. $y \in \Lambda (\G )$ iff there is an infinite
sequence of (different) elements $\gamma _n \in \G$ such that
$\lim \gamma _n x =  y$ ).

It is clear that both the domain of discontinuity and the limit set are $\Ga$-invariant.

\begin{rem}
More generally one can define the limit set for each subgroup $\Ga
\subset Isom(\H^n)$ as follows. Start with a point $x\in \H^n$,
then consider the closure $cl(\Ga x)$  of the $\Ga$-orbit of $x$
in $\H^n\cup \D \H^n$. Finally let $\La(\Ga):= cl(\Ga x) \cap \D
\H^n$.
\end{rem}

The Convergence Property implies that if $x \in \Omega (\G )$ then
$z_x \notin cl_{\hat{G}}(\G )$ for any $z$.
It follows that $\Lambda (\G )$ does not depend on the choice
of $x \in \Omega (\G )$. Also:  $\Lambda (\G ) \cap \Omega (\G )
= \emptyset$.

\begin{thm}
 $\c = \Lambda (\G ) \cup \Omega (\G )$.
\end{thm}
\proof Let $x \notin \Omega (\G )$. Then there exist a sequence
$g_n \in \G$ and $z_n \to x$ such that
$$
\lim_{n\to \infty} (g_n ( z_n )) = x \eqno(54)$$
 Then
up to subsequence $g_n \to a_b$. Since $x \ne b$ then (54) implies
that $x= a$ which is impossible. \qed

\medskip
{\bf Example:}
 For each elementary Kleinian group the limit set consists of
$0, 1$ or  $2$ points.

\begin{lem}
For nonelementary groups $\Lambda (\G ) = cl( \G x )$ for each
$x \in  \Lambda (\G )$.
\end{lem}
\proof Let $x, z\in \La(\Ga)$. There exists a sequence $g_n\in
\Ga$ such that $g_n(p)\to z$ for all $p\in \Om(\Ga)$. By taking a
subsequence (if necessary) we can assume that $g_n \to z_w$. If
$w\ne x$ then $g_n(x)\to z$ and  we are done. Otherwise find $f\in
\Ga$ such that $f(x)\ne x$. Then $g_n f(x)\to z$. \qed

\begin{cor}
If $\Ga$ is nonelementary then $\La(\Ga)$ is the smallest nonempty
$\Ga$-inva\-riant closed subset of $\bar\C$.
\end{cor}

\begin{thm}
 If $\G$ is not elementary, then the loxodromic fixed points
are dense in  $\Lambda (\G )$.
\end{thm}
\proof First we need to find a loxodromic element in $\Ga$.
Indeed, the group $\Ga$ is infinite, hence it contains either a
parabolic or loxodromic element $g$. If $g$ is a  loxodromic
element we are done. Suppose that $g$ is parabolic. The fixed
point $p$ of $g$ is not fixed by the whole $\Ga$, hence there is
$h\in \Ga$ such that $h(p)=q\ne p$. Then the element $f= h g
h^{-1}$ is again parabolic and its fixed point is $q$. By
conjugating $\Ga$ in $PSL(2,\C)$ we can assume that $g : z \mapsto
z + a$. Since $g$ is parabolic there are two closed tangent (at
$p$) round discs $D, D' \in \C$ such that $f (int D) = ext (D '
)$. Then, for large $n$ we have: $E= g^{-n} (D) \cap D' =
\emptyset$. It follows that $\al:=f \circ g^n : int E \to ext D'$.
Thus, the  iterations of $\al$ show that $\al$ is loxodromic and
it has a fixed point $x\in \La(\Ga)$. Finally, by the previous
lemma the $\Ga$-orbit of $x$ is dense in $\La(\Ga)$. For each
$\ga\in \Ga$ the fixed-point set of $\ga \al \ga^{-1}$ is the
$\ga$-image of the fixed point set of $\al$. Hence $\Ga x$
consists of fixed points of loxodromic elements of $\Ga$.  \qed

\begin{cor}
 For nonelementary group, loxodromic fixed pairs are dense in
$\Lambda (\G ) \times \Lambda (\G )$.
\end{cor}
\proof Take disjoint open $U$ and $V$ which intersect the limit set.
Then there are loxodromic $p, q$ such that $p^n \to x \in U$ , $q^n \to y \in V$.
Find a lox element $f$ with fixed points different from that of $p$ and
take $g = p^n f p^{-m}$. Then fixed point of $g$ are in $U$ and
 $g^n \to z_w$ where $z, w \in U$. Thus, for some large $k$ we have:
$h_n = q^k g^n \to v_w$ where $v \in V$. Thus, one fixed point of
$h_n$ is in $U$, another is in $V$. \qed

So, if $\G$ isn't elementary and Kleinian , then the limit
set of $G$ is perfect, closed and has empty interior.

\section{Fundamental domains and quotient-surfaces}

Let $\Ga \subset \sll$ be a Kleinian group. A subset $F \subset \Omega (\Ga)$ is called {\it a fundamental set} of the group $\Ga$ if:

(i) $gF \cap F = \emptyset$ for all $g \in \G - 1$ with the exception:
$gF \cap F$ is a fixed point (or the set of two fixed points) of an elliptic
$g\in \Ga$, and

(ii)
$$
 \G F  \equiv \bigcup_{g \in \G} gF = \Omega (\G) \eqno(55)$$

\medskip
This is reasonable but too general definition. It allows to reconstruct the
 surface $S(\G) = \Omega (\G) /\G$ as a set, but we would like also
to recapture the topology of $S(\G)$ as well.

\begin{defn}
 A {\bf fundamental domain} $D$ for a Kleinian group $\Ga$
is an  open subset of $\Omega (\Ga)$ such that:

(1) ``The fundamentality'': There is a fundamental subset $F \subset cl (D)$, $D \subset F$ for
 the group $\Ga$;

(2) The ``side-pairing property'': The boundary of $D$ in  $\Omega (\G)$ is piecewise-smooth submanifold
in $\Omega (\G)$ and is divided in a union of smooth arcs which are called
{\bf sides} (or {\bf edges}\footnote{Their end-points are called the vertices}.); for each side $s$ there another  side $s'$ and an element
$g = g_{ss'}\in \G - 1$ so that $gs = s'$ ($g$ is called the ``side-pairing transformation''); $g_{ss'} = g_{s's} ^{-1}$.

(3) The ``finiteness condition'': The action of $\Ga$ defines an equivalence relation on the boundary of $D$ in $\Om(\Ga)$. We require the equivalence set of each vertex of $D$ to be
finite\footnote{This property is void if $D$ is compact}.
\end{defn}

Notice that the property (2) implies that $\D _{\Omega (\G)} D$ is
``locally finite'' in $\Omega (\G)$ , i.e. each compact $K \subset \Omega (\G)$
can intersect not more than finitely many sides of $D$.

\medskip
There are several other conditions on $D$ which imply (3):

\smallskip
(3') The orbit $G D$ is {\it locally finite} in
$\Omega (\G)$, i.e. for each compact $K \subset \Omega (\G)$ there is not
more than finitely elements $g \in \G$ such that $gD \cap K \ne \emptyset$.

\medskip
Alternatively: (3'') $D$ has only finitely many components.

\medskip
It is obvious that (3')$\Rightarrow$(3), the fact that (3'')$\Rightarrow$(3')
is less obvious, see Theorem \ref{locfin} below.

\medskip
Introduce the equivalence relation ``$\cong$''
 on $\D _{\Omega (\G)} D$ generated by the equivalence:
$x \cong y$ iff there is a
``side-pairing transformation'' $g$ such that $gx = y$. The factor-space
$$
E= cl_{\Omega (\G)}(D) /\cong$$
 has the quotient topology.
Denote by  $\pi $  the projection $\Omega (\G) \to \Omega (\G)/\G = S(\G)$.

\begin{thm}
 The natural map $\theta : E \to S(\G)$ is a
 homeomorphism, where $\theta: [x]\in E \mapsto Gx \in S(\G)$.
\end{thm}
\proof The projections $\tilde{\pi} : cl D \to E$ and $\pi$ are
open which implies that $\theta$ is continuous. The map $\theta$ is
surjective, thus we need to prove that $\theta$ is injective and open. It's
easy to see that the restriction of $\theta$ to the complement to the
projection of the set of vertices of $\D D$ is open and injective. Thus, our
problem is the set of vertices. Let $x \in \D _{\Omega (\G)} D$
 be any vertex, $x_1$ is the end point of a side $s_1$, then there is another
 side $s_1 '$ and the pairing element $g_1$ such that $g_1 (s_1 ) = s_1 '$,
$x_2 = g_1 (x_1 )$. Now, $x_2$ is the vertex for 2 sides: $s_1 '$
and another side $s_2$. For $s_2$ we again find a pairing
transformation $g_2$ and get $x_3 = g_2 (x_2 )$ etc. Thus, after
finitely many steps we end up with some point $x_n = x_{k}$, $k <
n$. In this case rename our sequence so that $x_1 = x_k , x_2 =
x_{k+ 1}$ etc. The product $h= g_1 ^{-1} \circ ... \circ g_{n-1}
^{-1}$ maps $x_1$ to $x_1$. Let $U_1$ be a small \nbd ~ of the
point $x_1$ in $cl D$, $U_2$ be small \nbd ~of  the point $x_2$ in
$cl D$ (see the  Figure \ref{fig:fund}) etc. Put $V = U_1 \cup g_1
^{-1} (U_2 ) \cup ... \cup g_1 ^{-1} \circ ... \circ g_{n-2} ^{-1}
U_{k-1}$ and the element $h$ maps the ``free'' boundary component
$s$ (of $V$) adjacent to $x_1$ and different  from $s_1$ to
another ``free'' boundary component $\sigma \subset g_1 ^{-1}
\circ ... \circ g_{n-2} ^{-1} U_{k-1}$. The element $h$ is a
priori  nontrivial since $G$ can have torsion. Put $W = V \cup h V
\cup ... \cup h^{q} V$, where $q+1$ is the order of $h$. Then $W$
is a \nbd ~ of $x_1$ and the images of $U_j$ cover it without
``overlaps''.


\begin{figure}[tbh]
\centerline{\epsfxsize=3in \epsfbox{fig2.eps}}
\caption{}
\label{fig:fund}
\end{figure}

This has two consequences. (1) If $g \in \G$ is such that $g x = y$, $x= x_1$
and $y \in \D D$, then $g^{-1} D \cap W$ should be
$h^i \circ g_1 ^{-1} \circ ... \circ g_{n-j} ^{-1} (D) \cap W$. Thus,
$g^{-1} =  h^i \circ g_1 ^{-1} \circ ... \circ g_{n-j} ^{-1} (D)$ since
$D$ is a fundamental domain and so, $x \cong y$.

(2) Let $A \subset E$ be open, then there is $B$ which is an open
 subset of $\Omega$ such that $cl D \cap B = \tilde{\pi}^{-1} (A)$. Let
$Z$ be the $G$--orbit of $cl D \cap B$. Then $Z$ is a \nbd of $x$
according to the discussion above. However, $\pi (Z) = \pi ( cl D
\cap B) = \theta \circ \tilde{\pi} ( cl D \cap B) = \theta (A)$.
Thus, since $\tilde{\pi}$ is open, $\theta (A)$ is a \nbd ~of
$\theta (\pi (x))$. This is true for all vertices of $\D D$. Thus,
$\theta$ is open. \qed


\begin{thm}
\label{locfin}
Suppose that $D$ satisfies (1) and (2) and has only
finitely many connected components. Then $D$ satisfies the
local finiteness condition (3').
\end{thm}
\proof Suppose that $\Omega \ni x \in \lim g_n D^0$, where $D^0$ is a
connected component of $D$. There can be not more than countably many exceptional
points $x \in \Omega (\G)$; all are vertices of $\D D$. If $D^0$ is
 relatively compact, then  we are done. If not, then there is a point
$z \in \D \cap \Lambda (\G)$. Let $w = \lim g_n z \in \Lambda (G)$.
Let $E$ be the set of points in $\c$ whose \nbd s intersect $g_n D^0$
infinitely
 many times (the set of ``exceptional'' points and the limit set are contained
in $E$) . Then (since $D^0$ is connected)
 $x$ and $w$ belong to a common connected component of $E$ which
is impossible. \qed

\begin{exe}
Let $\Ga$ be the cyclic group generated by $z\mapsto 2z$.
Construct example of an open connected domain $D\subset \Om(\Ga)$ which satisfies (1), has piecewise-smooth boundary, however $cl_{\Om(\Ga)}(D)/\Ga$ is not compact.
\end{exe}


\subsection{Dirichlet fundamental domain}

Let $\G$ be a discrete subgroup of $PSL(2, \C )$, suppose that
$O\in \H ^3 = X$ is not a
fixed point of any nontrivial element of $G$. Then, define
$D_g = \{ x \in X : d(x, O) < d(x, gO ) \}$ for  $g \in \G- 1$;
$\tilde{D}_g = cl D_g - \D _X D_g$ where the closure is taken in
$\overline{\H ^3} = \H^3 \cup \c$.
 Define  $B_g$ to be the hyperbolic plane $\D _X D_g$.

\begin{defn}
 The intersection
$$
D_O (\G) = \bigcap _{g \in \G- 1} D_g
$$
is called the {\it Dirichlet polyhedron of} $\G$ {\it with center at} $O$.
The set
$$
\Phi _O (\G) = \bigcap _{g \in G- 1} \tilde{D}_g \cap \c$$
is called the {\it Dirichlet fundamental domain for} $\G$.
\end{defn}

\begin{thm}
 $\Phi _O (\G) $ is a fundamental domain for the action of $\G$ in
$\Omega (\G) \subset \c$.
\end{thm}
\proof Let $x\in B_g \cap D$. Pick $h \ne g^{-1}$. Then $d(g^{-1}(x), h(O))=
d(x, g\circ h(O)) \ge d(x, O)= d(g^{-1}(x), O)$. This implies the ``side-pairing property''.
The nontrivial statement is that for each $z \in \Omega (\G)$
there is an element $g \in G$ such that $z \in cl(g(\Phi _O (G)) $.
If $z$ doesn't belong to $G$-orbit of the closure of any face
 of $D_O$ then the conclusion
follows from the fact that there is a \nbd \ $V$ of $x$ in $\H ^3$
such that $V \subset g(D_O )$; thus $z \in g(\Phi )$. Suppose
else. Then the set of such points form a nowhere dense subset $E$
in $\Omega (G)$. Consider $p: \Omega (G) \to \Omega (G)/G = S$ ;
$p(z) \in p(E)$, $p(z) = \lim p(z_n )$, $p(z_n ) \in S - p(E)$.
Thus, $z = \lim g_n (z_n )$, $z_n \in \Phi $. If $g_n$ is
relatively compact, then $g_n = g$, $z = \lim g(z_ n) \subset cl
g(\Phi )$. Otherwise, $g_n \to a_b$. However, $diam (g_n (\Phi ))
\to 0$ as $n \to \infty$. Thus, $\lim g_n (z_n ) = \lim g_n (\Phi
) = z$ which is impossible. \qed

\begin{figure}[tbh]
\centerline{\epsfxsize=3in \epsfbox{fig3.eps}}
\caption{}
\label{fig:mod}
\end{figure}


{\bf Example.} Let $G$ be the {\it classical} or {\it elliptic}
modular group $SL(2, \Z )$ (later we shall deal also with
``Teichm\"uller modular group). Denote by $P \subset \H ^2$ be the
open triangle bounded by $\{ Re(z) =  \pm 1/2 \}$ and $\{ |z| =
1\}$. Then, $P$ is the Dirichlet polygon with center at $w= iv$,
$1 < v \in \R$. Really,

(i) $f(z) =  z +1 $ and $g(z) = - z^{-1}$ are in $G$. Then $P \subset D_w$.
Thus, we need only to show that $P$ has no equivalent points. Suppose that
$z \in  P , hz \in P$;
$$
h(x) = \frac{ax + b}{cx + d}$$
Then $|cz + d|^2 = c^2 |z|^2 + 2Re(z) cd + d^2 > c^2 + d^2 - |cd| =
(|c| - |d|)^2 + |cd| = \alpha$. Then $\a \in Z$ and $\a = 0$ iff
$c = d= 0$, thus $\a \ge  1$ and  $|cz + d| > 1$. Then,
$$
Im (hz) =  \frac{Im(z)}{ |cz + d|^2} < Im (z)$$ On another hand,
we have $hz \in P$, so $Im (hz) > Im(z)$. This contradiction shows
the absence of such point $z$. \qed

To be more precise, the Dirichlet fundamental domain for the action of
$G$ in $\c$ is the union of $P$ and it's image under inversion in $\D \H ^2$.
The center $O$ for this domain is contained in a copy of the hyperbolic
plane in $\H^3$ which is invariant under $G$, $O$ is a point on the geodesic
$L$ in $\H^3$ which connects the fixed points of $f$ and $g f g$ and lies
 between
a fixed point of $f$ and the fixed point $j$ for the action of $g$ on $L$.

Thus, $D_O (G)$ is bounded by 3 hyperbolic planes, two of them are tangent at
the fixed point of $f$. See Figure \ref{fig:mod}.



\bigskip


\subsection{Ford fundamental domain}


Another example of the fundamental domain is so called {\it Ford fundamental
domain}. Let $\sll \ni g : z \mapsto (az +b)/(cz +d)$, $g(\infty ) \ne
\infty$, i.e. $c \ne 0$; we assume that $ad - bc = 1$.
 Then, $g'(z) = (cz + d)^{-2}$ and define
$I_g$ to be the set of points in $\c$ where $g'$ is an Euclidean isometry
i.e. $I_g = \{ z \in \C : |g'(z)|  = 1 \} = \{ z \in \C : |cz + d| = 1 \}$.
 Then $I_g$ is a circle
which is called {\it isometric circle} of $g$. The center of this
circle is $g^{-1} (\infty ) = -d/c$ and $|c|^{-1}$ is the radius
of $I_g$. Thus, the radius of $I_g$ is the same as the radius of
$I_{g^{-1}}$. Any Euclidean circle with the center at $g^{-1}
(\infty )$ is mapped by $g$ in a Euclidean circle with the center
at $g (\infty )$ (since the last bunch of circles is described by
the property that they are orthogonal to each line through $\infty
, g (\infty )$).

However, the radius of $I_g$ should be the same as the radius of $g(I_g )$,
thus $g(I_g ) = I_{g^{-1}}$ and $g(ext I_g ) = int I_{g^{-1}}$. Now,
assume that $\infty \in \Omega (\G )$ and it isn't a fixed point of any
nontrivial element of the Kleinian group $\G$, then
$$
F(\G ) = \{ z \in \C : |g'(z)| < 1 , g \in \G \} =
\bigcap_{g \in \G - 1} ext I_g$$
is called the {\it Ford fundamental domain} of $\G$.

It's possible to realize $F(\G )$ as the degenerate case of the Dirichlet
fundamental domain. To do this we need we need another point of view on
$B_g$. namely, if $O \in \H^3$ then there is a unique ball $B(O, r)$
with center at $O$ (and radius $r$) such that $\D B(O, r)$ is tangent
to $\D (gB(O, r))$ at some
point $x = x_g \in \H^3$. Then $B_g$ is the unique hyperbolic
 plane which is tangent
to both $\D B(O, r)$, $\D gB(O, r)$ at $x$. Now we can move the point $O$
to the point $s$ at the ``infinity'' $\c$ of the hyperbolic space assuming that
$s$ isn't a fixed point of $g$), thus $r \to \infty$ and
$B(O, r)$ degenerates to a horoball $U_g$ with center at $s$ and $ gB(O, r)$
-- to a horoball $gU_g$ with center at $gs$. (See Figure 4 below where
$s =  \infty \in \c$).

\begin{figure}[tbh]
\centerline{\epsfxsize=3in \epsfbox{fig4.eps}}
\caption{}
%\label{fig:cyclic m-fold covering}
\end{figure}



Then it's clear that $B_g \cap \c$ is the isometric circle of $g^{-1}$ since
it has  center at $g(\infty )$, the same (Euclidean) radius as $B_{g^{-1}}$ and
$g (B_{g^{-1}}) = B_g$. Notice also that the Euclidean diameters of
$I_{g(n)}$ tend to $0$ for  any infinite sequence of different elements
$g(n) \in \G$. Really, the centers of $I_{g(n)}$ belong to a compact subset
$K$ of $\C$ (since $\infty \in \Omega (\G )$) and the Euclidean distances
of the points $x_n = B_g \cap U_g$ to $\C$ are bounded from above
(since  $\infty \in \Omega (\G )$ and it isn't a fixed point of any
 element of $\G -1$). Thus, either infinitely many points
$x_n$ belong to a compact subset of $\H^3$ (which contradicts to the
 discreteness of $\G$) or $dist (x_n , \C )= Rad (I_{g(n)}) \to 0$. The
last implies that we can apply to $F(\G )$ the same arguments as to
$\Phi _O (\G )$ to prove that it is fundamental.



\subsection{Quasiconformal  conjugations of Fuchsian groups}
\begin{thm}
 Suppose that $S$ is a hyperbolic surface of finite area;
$S = \H ^2 /G$. Then $\Lambda (G) = \S ^1 = \D \H ^2$.
\end{thm}
\proof Suppose that $x \in  \S ^1 \cap \Omega (G)$ , then let $V_x
= U_x \cap \H ^2$, so that $gV_x \cap V_x = \emptyset$ for all $g
\in G - 1$. Then, $V_x$ projects isometrically to $S$; thus
$\infty = Area( V_x ) < Area(S)$. \qed

\begin{cor}
Fixed points of loxodromic elements of $G$ are dense on
$\S ^1$.
\end{cor}

\begin{thm}
 (Extension Theorem). Each \q.c. self-map of
the unit disc $\Delta$ can be extended continuously
to $ext(f) :
 \D \Delta \to \D \Delta$ (see Property 6 of \q.c. maps).
\end{thm}


Moreover suppose that $f : \Delta \to \Delta$ is a quasiconformal
homeomorphism such that $fgf^{-1} \in \sll$ for all elements of
some Kleinian group $\G \subset PSL(2, \R )$. Then we can extend
$f$ to a \q.c. map of $\c$ which conjugate $\G$ to a Kleinian
group. To prove this, extend the complex dilatation $\m$ of $f$ to
the Beltrami differential $\n$ on $\c$ using $j_* \mu$ (see (49))
where $j$ is the inversion in $\D \Delta$. Then, the solution of
the Beltrami equation $\BD \tilde{f} = \n \D \tilde{f}$ (after
composition with some Moebius transformation) defines the desired
extension of $f$ (since $\tilde{f}$ is a self-map of $\Delta$, the
solution of the Beltrami equation in $\Delta$ is unique up to
composition with a conformal automorphism). Notice that the
coefficient of the quasiconformality of this extension is the same
as that of $f$.

\begin{thm}
 Suppose that $f_0 , f_1 :  X \to Y$.
Then $f_0$ is homotopic to $f_1$ they induce ``equivalent'' isomorphisms of
the fundamental group.
\end{thm}

\proof The nontrivial implication is $\Leftarrow$. The isomorphisms
are ``equivalent'' iff they differ by ``tale'' between $f_0 (x)$, $f_1 (x)$
where $x$ is the base-point. Thus, there are lifts
$\tilde {f}_j : \H ^2 \to \H ^2$ such that the induced isomorphisms
of $\pi _1 (X) , \pi _1 (Y)$ are equal to $\theta$;
i.e. $\theta (g) \circ \tilde {f}_j =  \tilde {f}_j \circ g$ for all
$g \in  G = \pi _1 (X)$. Define $\tilde {f} (z, t)$ to be the
point of $[f_0 (z), f_1 (z)]$ which divides this segment as $t: (1-t)$.
Consider $\tilde {f} (gz, t) =$ the
point of $[f_0 (gz), f_1 (gz)]$ which divides
 as $t: (1-t)$ ; however, $[f_0 (gz), f_1 (gz)] =
[\theta(g) f_0 (z), \theta(g) f_1 (z)]$, so $\tilde {f} (gz, t) =$
the point of $[\theta(g) f_0 (z), \theta(g) f_1 (z)]$ which
divides as $t: (1-t)$. I.e. this is the same point as
$\theta(g)\tilde {f} (z, t)$. Thus, $\tilde {f} (z, t)$ projects
to the homotopy $X \to Y$ between $f_0 , f_1$. \qed

\begin{cor}
 Quasiconformal homeomorphisms
$f_0 , f_1 :  X \to Y$ are homotopic iff the extensions of $\tilde {f}_j$
to $\D \H ^2$ coincide (for some choice of the lifts).
\end{cor}

\begin{cor}
 Suppose that $f: X \to Y$ is a conformal automorphism
homotopic to $id$. Then $f = 1$.
\end{cor}

\begin{thm}
(Theorem of Nielsen). Let $X, Y$ be two surfaces
 of the same finite conformal type.
Suppose that $\phi : \pi _1 (X) \to \pi _1 (Y)$ is an  isomorphism,
such that the image of ``peripheral'' (representing a loop around a puncture)
element of $ \pi _1 (X)$ is again
 peripheral.
 homeomorphism. Then there exists
a  homeomorphisms $f :  X \to Y$ which induces $\phi$. If $h$ is orientation-
preserving, then $f$ is homotopic to a \q.c. homeomorphism.
\end{thm}

\begin{cor}
 Suppose that $G_1 , G_2$ be a pair of torsion-free
 Kleinian subgroups of
$PSL(2, \R )$ such that $\2h  /G _j$ have finite type and
$\theta : G_1 \to G_2$
is a {\it type preserving} isomorphism (i.e. the image of parabolic element is
parabolic etc.) Then, $\theta$ can be induced by a homeomorphism $\tilde{f}$ of
 the hyperbolic plane. If $\tilde{f}$ is orientation-preserving, then it
can be chosen \q.c. .
\end{cor}

We postpone the proof of this result. The isomorphisms that can be induced by
\q.c. maps are called {\it admissible}.


\subsection{Finiteness of area versus finiteness of type}

\begin{thm}

\end{thm}




\section{Teichm\"uller theory}
\subsection{Teichm\"uller space}
Now, let's go back to Riemann surfaces. Our current goal-
construction of Teichm\"uller space.
 Fix once and for all a \Rs \ $S$ of finite type.
The \q.c. maps $f: S \to Y , g: S \to Z$ are equivalent if $f
\circ g^{-1}$ is homotopic to a conformal map between $Z$ and $Y$.
The space of equivalence classes of the pairs $(Y, g)$ is the {\it
Teichm\"uller space} $T(S)$.

Several alternative definitions.

(i) Recall that each Riemannian metric on $S$
defines a complex structure. Consider the space $R(S)$ of Riemannian
metrics which
have finite conformal type.
The groups $Diff_0$
(of diffeomorphisms of $S$ homotopic to identity) and
$C^{\infty} (S, \R _+ )$ act on  $R(S)$ as $g \cdot ds = g^* ds$, and
$C^{\infty} (S, \R _+ ) \ni \phi : ds \mapsto \phi ds$.
 Then
$$
R(S)/Diff_0 \times C^{\infty} (S, \R _+ ) \cong T(S)$$
 Namely, if $[X, g] \in T(S)$ then
 $g^{-1} (X)$ is a complex structure on $S$ which corresponds
conformally to a
Riemannian metric of finite type on $S$. Conversely, if $ds$ is a metric,
let $f \in Diff_0 $ be so that $f^* ds$ near the punctures of $S$ is
conformally equivalent to the $S$.
Then define the Beltrami differential $\m$ as in the section 3.
Then define the Beltrami differential $\m$ as in the section 3.
The supremum norm of $\m$ is less than 1 and we can
 solve the Beltrami equation on $S$: $\BD h = \m D h$, $h: S \to S$.
then $(h(S, ds), h)$ projects to a point in $T(S)$.

If we consider the space
$H(S)$ of  complete metrics of
constant curvature $-1$ then the quotient $H(S)/Diff_0 = F(S)$ is called
``Fricke space'' and it's diffeomorphic to $T(S)$.

(ii) Using the Uniformization identify $\pi _1 (S) $ with a
discrete subgroup
$$
 F \subset  PSL(2, \R )$$
Consider the space $Hom_a (F \to PSL(2, \R ))$ consisting
of {\it admissible} monomorphisms (which have discrete images,
preserve  the type of elements and ``orientation''). Then take
the quotient $T(F) = Hom_a (F \to PSL(2, \R ))/ PSL(2, \R )$, where
$\gamma \in PSL(2, \R )$ acts on $r \in Hom_a (F \to PSL(2, \R ))$ by
 conjugation $\gamma \cdot r (g) = \g r(g) \g ^{-1}$, $g \in F$. This space
is called the Teichm\"uller space of the group $F$.

The space $T(F)$ has a natural topology induced by the matrix topology of the
group $ PSL(2, \R )$.

This space coincide with
$T(F) = \{ h: \Delta \to \Delta = \2h : h_* (g) \equiv h g h^{-1}
\in PSL(2, \R )$ for all
$g \in F$ , $h$ is quasiconformal $\}/\cong$, where $h_1 \cong h_2$ iff
there exists an element $\gamma \in PSL(2, \R )$ such that:
$$
ext (h_1 )|_{\D \Delta} = \gamma \circ ext (h_2 )|_{\D \Delta}$$
We can avoid the composition with elements of  $PSL(2, \R )$
assuming that all \q.c. \ maps $ext(h)$ fix 3 distinguished points $p_j$ on
$\D   \Delta$ (i.e. $ext(h)$ is normalized).

{\bf Teichm\"uller metric on} $T(S)$. If $p, q \in T(S)$ then put
$$
d_T (p, q) =
inf\{ \log K(f\circ g^{-1}) : f \in p , g \in q\} \eqno(56)
$$
The Convergence property for \q.c. mappings implies that the
Teichm\"uller distance is always achieved by some \q.c. map.
 The triangle inequality is
obvious since $K(f \circ g ) \le K(f) K(g)$;
$d_T (p, q) = d_T (q, p)$ since $K(h) = K(h^{-1})$.

Several other definitions of $d_T (p, q)$. Let $\tau _1 (p, q) =
\log inf \{  K( f\circ g^{-1}_0 ) : f \in [q]) \}$;  $\tau _2 (p, q) =
\log inf \{  K(h) : h \in [f\circ g^{-1}] \}$. Then:
$\tau _2 \le d_T \le \tau _1$; given $h$ we can take $g = h^{-1} f$, thus
$\tau _1 \le \tau _2$.

Another point of view. Recall that for $z, w \in \Delta$ the hyperbolic
distance
$$
d(z, w) = \log \frac{1 + |z - w|/|1- \bar{z}w |}
{1 -  |z - w|/|1- \bar{z}w |} = \log \frac{|1- \bar{z}w | + |z- w|}
{|1- \bar{z}w | - |z- w|} \eqno(57)$$
where $ds = \frac{|dz|}{1- |z|^2} \cong \frac{|dz|}{Im(z)}$.
Therefore (by (9)),
$$
d_T (p, q) = inf \log \frac{1 + \|\m - \n \|/\|1- \bar{\m}\n \|}{1
-
 \|\m - \n \|/\|1- \bar{\m }\n \|} =$$
$$
inf \{ d(\mu (z) , \nu (z)) : z \in \Delta , \mu \in [p], \nu \in [q] \}$$

\begin{rem}
\label{quiv}
 Let's compare two distances in the open unit ball $B(1)$
of complex characteristics. We have:
$$
\| \mu (f^{\m _1} \circ f^{\m 2} )\| = \|\m _1 -  \m _2 \|/\|1 -
\bar{\m _1} \m_2\| < 2 \|\m _1 -  \m _2 \|
$$
since $\|1 - \bar{\m_1} \m_2\| < 2$. On another hand, if $ \|\m
_1\|$ or   $\|\m _2\| \le C < 1$ then
$$
\|\mu (f^{\m _1} \circ f^{\m 2} )\| \ge  \|\m _1 -  \m _2 \| 2/(C
+1).$$
 Thus, for each $r < 1$ the metric on  $B(r)$ defined by the
supremum norm and the ``Teichm\"uller metric'' are equivalent.
\end{rem}

\medskip
Thus $d_T$ is a metric. The space $T(S)$ is path-connected.
If $X, Y$ are \q.c. equivalent then $T(X), T(Y)$ are canonically
isomorphic.

\subsection{The modular group}

Define the {\it (Teichm\"uller) modular group} $Mod_S$ as
 the quotient
$$Homeo_+(S)/Homeo_0(S)$$
where $Homeo _+ (S)$ is the group of ori\-entation pre\-ser\-ving
ho\-meo\-mor\-phisms of $S$ and $Homeo_0(S)$ is the subgroup of
homeomorphisms homotopic to $id_S$. Then  $Mod_S$ acts on $T(S)$
by the precomposition $Mod_S \ni g: [X, f] \to [X, f\circ g ]$.
The quotient $T(S)/Mod_S$ is called the {\it moduli space} of
complex structures on $S$. The  group $Mod_S$ is isomorphic to
$$
Out _+ (S) = Aut_+ (S) /Inn(S)$$
 where $Aut_+ (S)$ consists of
``admissible'' automorphisms ``preserving the orientation''.

The bijection $T(S) \to T(F)$ is continuous (with the
Teichm\"uller
 topology on $T(S)$
and the ``matrix'' topology on $T(F)$) because of the continuous
dependence of the solution of Beltrami equation on the dilatation. The fact
that this an open map is more subtle. One way to prove it is to show that
both are manifolds (as we shall see). Another way  is to prove so called
``quasiconformal stability'' which is more general fact. The rough idea of
the second approach is that small deformation of the representation leads
to small deformation of the Dirichlet fundamental polygon and we can
organize a \q.c. diffeomorphism of the fundamental domains which is $C^1$
close to $id$ and preserves the equivalence relation on the boundary
of domains.

\subsection{Teichm\"uller space of the torus}

Let $\G \subset Isom (\C )$ be a torsion-free lattice, $\G = <g_1 , g_2 >
\cong \Z ^2$. Using conformal
conjugations of  representations $r: \G \to Isom (\C )$ we can assume that
 $g_1 (z) = z$ and $g_2 (z) = z + \tau$. Thus, $\tau \in \{ t \in \C :
Re (t) > 0 \}$ is the only invariant of the conjugacy class of the
 admissible representation. Another way to describe this invariant is to
embed $Hom_a (\G ,  Isom (\C ) )$ in $\C ^2$ so that $r \mapsto
(t_1 , t_2 )$, where $r(g_j ) : z \mapsto z + t_2$. Then, the conformal
factorization of the space of representations is equivalent to
 the projectivization of $\C ^2$. The point $(t_1 : t_2 ) \in CP(1)$
corresponds to $\tau = t_2 /t_1$. Anyway, the Teichm\"uller space
of the torus $T(T^2 )$ is the upper-half plane $\2h$. The modular
group of the torus $Mod_{T^2}$ acts on $T(T^2 ) \subset CP(1)$
 as $SL(2, \Z ) = Aut _+ (S)$. This action isn't effective, so let
$PSL(2, \Z ) = Aut _+ (S) / \{ \pm I \}$.

{\bf Remark.} The same happens with the surfaces
of genus 2 which are hyperelliptic
and the hyperelliptic involution is always induced by conformal map of the
surface. But, anyway, the kernel is $\Z _2$.
Now the action of $PSL(2, \Z )$ on $\2h$ is just the action of the
``classical'' modular group.

The next step is to calculate the Teichm\"uller distance, in
particular to show the equivalence of the Teichm\"uller topology
with the topology of $\C$.
 Suppose that
$r_1 , r_2$ are admissible representation with the normalization
$r_j (g_1 ) = 1$; $r_1 (g_2 ): \tau \mapsto \tau + w$,
$r_2 (g_2 ): \tau \mapsto \tau + z$, $\tau \in \C$.

\begin{lem}
 There is an affine map of $\C$ which is an
 extremal quasiconformal map conjugating
$r_1$ and $r_2$  (it's a particular case of the Teichm\"uller's
analysis of the extremal maps which claims that moreover, the
extremal map is unique).
\end{lem}
\proof Let $L$ be the lattice
$\Z + w \Z \subset \C$, $L '$ be the lattice $\Z + z \Z \subset \C$. denote by
$A$ the unique $\R$-linear map which transforms $L$ to $L'$, $A(1) = 1$,
$A(w) = z$,
$$
A(\tau ) = \frac{z - \bar{w}}{w -  \bar{w}} \cdot \tau +
 \frac{w - z }{w -  \bar{w}} \cdot \bar{\tau} \eqno(58)$$
$$
\m (A) =  \frac{w- z}{z - \bar{w}} \eqno(59)$$
 Suppose that $f$ is an extremal map. After normalization we can
assume that $f(0) = 0$, $f(1) = 1$, thus $f(w) = z$. Therefore, restriction
of $f$ to $L$ coincides with $A$. Denote by $\Phi$ the rectangle
bounded by the segments $[0, 1], [1, w +1], [w+1 , w], [w, 0]$,
which is a fundamental domain for $\Gamma$.
For $k \in \Z _+$ put $f_k (z) = f(2^k z ) \cdot 2^{-k}$. Then
 $K(f_k ) = K(f)$ and they induce the same conjugation between
$r_1$ and $r_2$. However, the restriction of $f_k$ to $ 2^{-k}\cdot L$
coincide with $A$ and $L_k =  \Phi \cap 2^{-k}\cdot L \subset
L_{k+1}$. Therefore,
$$
L_{\infty} = \bigcup_{k \in  \Z _+} L_k $$
is dense in $\Phi$. On another hand, there is a uniform limit
$f_{\infty} = \lim f_k$ and the restriction of $f_{\infty}$ to $L_{\infty}$
 coincides with $A$. Thus, $f_{\infty} = A$ on $\C$, $K(f_{\infty} ) = K(f)$
and thus $A$ is an extremal map. \qed

Thus we can calculate $K(A)$ as
$$
K(A) = \frac{|z - \bar{w}| + | z - w|}{ |z - \bar{w}| - | z - w|} \eqno(60)$$
and $d_T (r_1 , r_2 ) = log K(A)$ is the hyperbolic distance between
$z$ and $w$. Really, on the imaginary axis it coincides with $\log {w/z}$
if $Im(w) \ge Im(z)$. The invariance of $K(A)$ under homotheties
and translations in $PSL(2, \R )$ is evident. The invariance under the
inversion $j: \tau \mapsto 1/\bar{\tau}$ can be verified by a
direct calculation.

 Thus, in the case of the torus the Teichm\"uller and hyperbolic
metrics coincide. This is a particular case of more general theorem
of Royden that we shall (probably) discuss later.

{\bf Remark.} Another simple case is the punctured torus $S_{1,1}$.
 In this case the
 Teichm\"uller space is the same as the Teichm\"uller space for the torus.
Really, let $[X, f] \in T( S_{1,1})$, $f$ is quasiconformal.
 Then $\phi ([X, f]) \in T(T^2 )$
is just the extension of the complex structure  and
of the quasiconformal map to the puncture. This map is injective since the
group of conformal automorphisms of the torus is transitive.

\medskip
\subsection{Simple example of the moduli space.}

 Let $S$ be a sphere with
$n+3$ punctures. Then the moduli space of $S$ is
$$
(\C - \{0, 1 \} )^{n}- \hbox{diagonals} /S(n)$$ where $S(n)$ is
the symmetric group on $n$ symbols. The fact that the topology is
the same as Teichm\"uller's just follows from the continuous
dependence of solution of Beltrami equation on the dilatation and
the fact that if $(z_1 ,..., z_n ) \in \C ^n$ is close to $(w_1
,..., w_n ) \in \C ^n$ then there is a diffeomorphism $f: \c \to
\c$ which is $C^1$ close to $id$ such that $f(z_j ) = w_j$.

\subsection{Completeness of the Teichm\"uller space.}

\begin{thm}
 The space $T(S)$ is complete.
\end{thm}
\proof  Let $[X_n ,f_n]$ be a Cauchy sequence in $T(S)$.
$f_n : \c \to \c$ be a Cauchy sequence in $T(S)$. Without loss of generality
we can assume that  $X_n$ are complex structures on one and the same
smooth surface $S$.
First fix $f_i$ such that
$$
inf \{ \log K(f_{i+p} \circ f_i ^{-1}) ,
f_{i+p} \in [f_{i+p}]\} < 1/2$$
 for all
$p =1, 2,...$. Renumber the sequence, put $f_i = f_1$. Then choose $f_2$
such that $\log K_{f_{2} \circ f_1} < 1/2$ and $d_T ([X_i , f_i ] ,
[X_{i+p} , f_{i+p} ] \le 1/4$. Finally we get a sequence of maps $f_n : S \to X_n$ so that
$$
\log  K_{f_{n+1} \circ f_n ^{-1}} < 2^{-n}$$
for each $n$. Then this
 Cauchy sequence in $T(S)$ has  $\log  K_{f_{n+p} \circ f_n} < 2^{-n +1}$
for each $p$. Thus,
$$
\| \m _{n+p} - \m _n\|_{\infty} \le 2\|( \m _{n+p} - \m _n )/ (1-
\bar{\m}_n \m _{n+p} )\| < 2 \exp \frac{2^{-n+1 } -1}{2^{-n+1 } +
1} = 2 \tanh 2^{-n +1}
$$
(See Remark \ref{quiv}) Thus $\m _n$ is a Cauchy sequence in
$L_{\infty}$. Since $L_{\infty}$ is complete, the limit $\m$
exists. On another hand, $K(f_n) \le K(f_1)2^{-n+1 } < C <
\infty$. This estimate also implies that $\|\m \| < 1$ and we can
solve the Beltrami equation with $\m$. The solution- $f^\m$
belongs to $T(S)$. The points $\m _n$ are convergent to $\m$ in
the supremum norm, thus the points of the Teichm\"uller space are
convergent in the Teichm\"uller topology.
 \qed

\subsection{Real-analytic model of the Teichm\"uller space}

We will also use another realization for $T(S)$.

Namely, let $\G$ be the Fuchsian group such that $\H ^2 /\G = S$.
Then consider the space $B(\G )$ of all Beltrami differentials
which are automorphic under $\G$:
$$
B(\G ) = \{ \mu \in L_{\infty} (\2h ) : \|\m \| < 1, \m (z) = \m
(\g z) \overline{\g '(z)}/\g '(z) , \g \in \G \} \eqno(61)
$$
Introduce the equivalence relation: $\m \sim \nu$ iff $f^\m $ and $f^{\nu}$
coincide on $\D \H ^2$ (q.c. maps are canonically normalized and the
complex dilatation is extended to $\c - \2h $
by the inversion). Then $f$ conjugate Fuchsian groups to Fuchsian groups.
Denote by $\Psi : B (\G ) \to T(S)$ the corresponding projection.

\subsection{Complex-analytic model of the Teichm\"uller space}

 We now extend the complex dilatation $\m$  to $\c -\2h$ by $0$. Denote by $f^{\m}$ the normalized solution of the equation
$\BD f = \m \D f$ in $\c$.
Then (by formula (50)) we still have:
 $f^{\m} _* (g) = f^{\m} \circ g \circ (f^{\m})^{-1} \in \sll$, $g \in \G$.
Thus, $f^{\m}$ conjugates the group $\G$ to a Kleinian group $\G '
= f_* (\G )$. The group $\G '$ has two simply connected components
of the discontinuity domain- images of $\2h$ and $\2h _*$. Then
the surface $f^{\m} (\2h _* ) /\G '$ is conformally equivalent to
$\2h _*  /\G '$. The surface $f^{\m} (\2h  ) /\G '$ with the
marking given by the isomorphism of fundamental groups $f^{\m} _*
: \G \to \G '$ gives the point of the Teichm\"uller space $T(S)$.
This point is the same as the point of $T(\G )$ given by the
solution $f_{\m} : \2h \to \2h$ of  the Beltrami equation $\BD f =
\m \D f$ in $\2h$. Indeed, the map $g = f_{\m} \circ (f^{\m}
)^{-1}$ of the domain $f^{\m} (\2h  )$ is conformal (by uniqueness
of solution of Beltrami equation) and $g$ conjugates the
representations  $f^{\m} _*$ and  $f_{\m *}$.

Thus we have the correspondence: $\Phi : [\m ] \in T(X) \mapsto
 f|_{\2h _*}$ is a univalent (i.e. injective)
holomorphic function in $\2h _*$.

\begin{thm}
 The map $\Phi$ is injective.
\end{thm}
\proof If $f = g$ on $\2h _*$ then their extensions to $\R$ also
coincide. Thus, the induced homeomorphisms $f_*$, $g_*$  of the
group $\G$ to $\sll$ are the same and $f(\2h ) = g(\2h )$,
therefore the surfaces $f(\2h )/ f_*(\G)$, $ g(\2h ) / g_*(\G)$
are the same and their markings  defined by  $f_*$, $g_*$
coincide. \qed

Thus, we obtained an embedding of $T(X)$ in the space of holomorphic
 functions in $\2h _*$. Certainly, this map is very far from being surjective.
Our aim is to describe somehow the image.

\section{Schwarzian derivative and quadratic  differentials}

\subsection{Spaces of \qd s }

 Let $Q(\G )$ be the space of all holomorphic
in $\2h _*$ functions $\f$ such that $\f dz^2$ is $\G$-automorphic
(i.e. $\f (z) = f(\g z) \g ' (z) ^2$)
and have finite norm:
$$
\|\f \| = \sup_{z \in \H ^2 _*} y^2 |\f (z)| \eqno(62)$$ where we
realized $\2h _*$ as the lower half plane. The norm $\|\f \|$ is
invariant under the precomposition with elements of $\G$.

Notice that the condition (62) is equivalent to finiteness of
the $ds^2$-norm of the projection of $\f$ to $X =(S, ds^2) = \2h _* /G$.

Later we shall show that the space $Q(\G )$ is finite-dimensional.
 Its dimension is $3g - 3 + n$,
where $(g, n)$ is the type of $X$.

Schwarzian derivative of the holomorphic function $f$ is
$$
S_f \equiv \{ f, z\} = (\frac{f''}{f'})' - \frac{1}{2} (\frac{f''}{f'})^2
= \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2 \eqno(63)
$$
Suppose that $f \in \sll$, then $S_f \equiv 0$. The inverse statement also
holds for the holomorphic functions. Suppose that $S_f = \f$.
Put $h = \frac{f''}{f'}$,
then $h' - h^2 /2 = \f$, if
$h = - 2\eta '/\eta = -2 (\log \eta )'$ then for $\eta$ we have the
 Riccati equation:
$$
\eta '' + \f \eta /2 = 0$$
If $\f = 0$ then the only solution is: $\eta (z) = cz + d$.
However, $h (z) =  -2 (\log \eta )' =  \frac{-2c}{cz +d}$;
$g : = f'$ implies $(\log g )' = h$
$$
\log g = \int h = -2 \int \frac{cdz }{cz +d} = \log ((cz + d)^{-2}) + \log a$$
Thus, $f' = g = a/(cz +d)^2$,
$$
f = a \int  \frac{cdz }{(cz +d)^2} =  \frac{\a \b d + c\b z}{cz +d} \in \sll$$
{\bf Then} $S_f = 0$ {\bf iff} $f$ {\bf is Moebius.}
Under the composition the \Sd behaves as:
$$
S_{f\circ g} = (S_f \circ g){g'}^2 + S_g \eqno(64)$$
If $g$ is Moebius, then
$$
S_{f\circ g} =  (S_f \circ g){g'}^2$$
in particular, if $F(z) = f(1/z)$ then $z^4 S_F (z) = S_f (1/z)$.

Thus, $S_{f\circ g} = S_f$ for all $g \in \G$ iff
$S_f$ is $\G$-automorphic quadratic differential.

\begin{thm}
\label{schwarz}
 The (holomorphic) solution of the equation
$$
S_f = \f$$
exists and unique up to (left) composition with a Moebius transformation.
This solution is locally injective.
\end{thm}
\proof Using  the substitute $h = f'' /f$, $h = - 2\eta '/\eta$;
as above we have the Riccati equation:
$$
\eta '' + \f \eta /2 = 0 \eqno(65)$$
This equation has 2 linearly independent holomorphic solutions.
(Proof: power series expansion.) Let $\eta _1$, $\eta _2$ be such solutions.
Then $(\eta _1 \eta _2 ' - \eta _2 \eta _1 ')' =  \eta _1 \eta _2 '' -
\eta _2 \eta _1 '' = 0$. Thus, $\eta _1 \eta _2 ' - \eta _2 \eta _1 ' =
\hbox{const} \ne 0$ (otherwise, $\D (\log \eta _1 ) =  \D (\log \eta _2 )$,
 $\eta _1 = a \eta _2$). Thus we can put $\hbox{const} = 1$. The function
$f = \eta _1 / \eta _2$ satisfies the equation $S_f = \phi$.

Notice that $\eta _2$ can have at most simple zeros (since if $\eta _2 (z) =
\eta _2 '(z)$ then $\hbox{const} = 0$). Thus, $f$ has at most
 simple poles. In other points:
$$
f' = (\eta _1 \eta _2 ' -
\eta _2 \eta _1 ')/ \eta _2 ^2 = 1/ \eta _2 ^2 \ne 0 \eqno(66)$$
So, $f$ is locally univalent.

\medskip
{\bf Uniqueness of solution}. It follows from (64) that
$$
0 = S_{f \circ f^{-1}} \equiv (S_f \circ F^{-1})(f^{-1\prime})^2 +
S_{f^{-1}}$$
 and, since, $S_g = S_f$ we have:
 $$
 0 = (S_f \circ
F^{-1})(f^{-1\prime})^2 + S_{f^{-1}} = S_{g \circ  f^{-1}}.
$$
Thus, $g \circ  f^{-1} \in \sll$. \qed

\begin{thm}
 Suppose that the domain of $\f$ includes the point
$\infty$ and $\f$ has a simple pole at $\infty$. Then the solution of
$S_f = \f$ has at worst a simple  pole at  $\infty$.
\end{thm}
\proof  Suppose that $\f (w) = \psi (w) w^{-4}$, let
 $F(z) = f(1/z = w)$.  Then
$$
z^4 S_F (z) = w^{-4} \psi (w) = z^4 \psi (w) \eqno(67)$$ and we
have the equation $S_F (z) = \psi (1/z)$ where $\psi (1/z)$ is
holomorphic near $0$. Then we use Theorem \ref{schwarz} to
conclude that the function $F$ (and thus $f$) is meromorphic with
at worst simple poles. \qed

\begin{cor}
 Suppose that $\phi$ is $\G$-automorphic. Then the solution
of the equation $S_f = \f$ defines a homomorphism $\r : \G \to \sll$
such that $f \circ \g = \r (\g ) \circ f$ for each $\g \in \G$.
\end{cor}
\proof $S_{f\circ \g} = S_f$ and uniqueness implies that there is
$\r (\g )$ such that $f \circ \g = \r (\g ) \circ f$. \qed

\medskip
Thus, any automorphic \qd  \ defines a complex projective
structure on $\2h /\G$ which is ``subordinate'' to the initial
complex structure. And vice versa, given a holomorphic developing
map $d: \2h \to \c$ which is $\G$-equivariant, we have the
automorphic \qd  \ $S_d$. Our problem is to determine, which \qd
\ correspond to the points of the Teichm\"uller space.

\begin{thm}
(Kraus- Nehari)
 Suppose that $f: \2h _* \to \c$ is a univalent function.
Then
$$
| \{ f, z\} Im(z) ^2 | \le 3/2 \eqno(68)$$
\end{thm}
\proof Step 1. We will need the following

\begin{lem}
Let $F(\zeta )$ be a univalent holomorphic function in
$ext (\Delta ) = \{ |\zeta | > 1 \}$ so that $F(\infty ) = \infty$,
$F(\zeta ) = \zeta + b_1 /\zeta + b_2 /\zeta ^2 + ...$.  Then
$|b_1 | \le 1$.
\end{lem}
\proof Let $r > 1$;  $D_r$ be the complement to the image
$F( \{ |\zeta | \ge r \})$
then
$$
Area (D_r ) = \frac{1}{2i} \int_{ |\zeta | = r } \bar{F} dF > 0$$
$$
 \frac{1}{2i} \int_{ |\zeta | = r } \bar{F} dF =
 \frac{1}{2i} \int_{ |\zeta | = r } [\bar{\zeta} +
\bar{b}_2 /\bar{\zeta} ^2 + ...][1 -  b_1 /\zeta ^2 - 2 b_2 /\zeta ^3 - ...]
d\zeta$$
However,
$$
 \int_{ |\zeta | = r } \frac{d \zeta}{\zeta ^s \bar{\zeta}^p} = 0$$
if $s \ne p +1$ and $= 2\pi i r^{1-s -p} =  2\pi i r^{-2p}$ if
$s = p+1$. Thus, the hole integral is equal to:
$$
\pi [r^2 - \sum_{p=1}^{\infty} \frac{p|b_p|^2}{ r^{2p}} ] > 0$$ If
we put $r= 1$ then we will get in particular $|b_1| \le 1$ \qed

Step 2. Now we can estimate $\{F, \zeta \}$. We have:
$$
F' = 1 - b_1 /\zeta ^2 - ... \ \ \ F'' = 2 b_1 /\zeta ^3 + ... $$
$$
F''' = -6 b_1 /\zeta ^4 - ...$$
$$
S_F = F'''/F' - 3(F''/F')^2 /2 = (- 6 b_1 /\zeta ^4 - ...)(1 +
 b_1 /\zeta ^2 ... ) -  $$
$$
\frac{3}{2} (2 b_1 /\zeta ^3 + ...)^2(1 +
 b_1 /\zeta ^2 ... )^2 = -  6 b_1 /\zeta ^4 + O(1/\zeta ^6)$$

Step 3.  Finally, we consider the functions in the lower half-plane.
Let $x_0 + iy_0 = z_0 \in \2h _*$, then the transformation
$\zeta = \g (z) =(z- \bar{z}_0)/(z - z_0)$ maps $\2h _*$ to
the exterior of $\Delta$ and $\g (z_0 ) = \infty$. Then
put $f(\zeta ) = f( z (\zeta ))$; $f = F \circ \gamma$, thus
$$
\{ f, z \} = \{ F, \zeta \} \g ' (z) ^2$$
$$
 \g ' (z) = - \frac{2iy_0}{(z - z_0 )^2}$$
and
$$
\{ f, z \} = (-6b_1 + \hbox{powers of \ }1/\zeta )\zeta ^{-4}
\frac{-4y_0 ^2}{(z - z_0 )^4}
$$
$$
\zeta ^{-4} = \frac{(z- z_0 )^4}{(z- \bar{z}_0 )^4}$$
Therefore if $z = z _0 , \zeta =  \infty$ we are left with
$$
|\{f, z_0 \}| =  | 6b_1 \frac{1}{4y_0 ^2}| \le  \frac{3}{2 y_0
^2}$$ Theorem is proved. \qed

\begin{thm}
Suppose that $\|\f \| \le 1/2$, then the solution of the equation
$S_f = \f$ admits a \q.c. extension to $\c$ which is
$\r$-equivariant where $\r$ is the monodromy of the complex
structure as above. The complex characteristic of this extension
depends continuously on the norm of $\f$.
\end{thm}
\proof  We will not only establish the existence of the extension but
also construct the {\it canonical} one.
 Let $\eta _j$ be 2 linearly independent
  solutions of the Riccati equation $\eta '' = - \f \eta /2$.
Put
$$
\hat{f}(z)  = \frac{\eta _1 ( \bar{z}) + (z - \bar{z})\eta _1 ' (\bar{z})}
{\eta _2 (\bar{z}) + (z - \bar{z})\eta _2 ' (\bar{z})}= Q_1 (z) /Q_2 (z)$$
for $z \in \2h$ and
$\hat{f}(z) = \eta _1 (z) / \eta _2 (z) = f(z)$
 for $z \in \2h _*$. Then the function $\hat{f}(z)$ is smooth in its
domain and  $\BD \hat{f} / \D \hat{f} (z) = -2 y^2 \f (\bar{z}) =\m$
 in $\2h$ and $\BD  \hat{f} \equiv 0$ in $\2h _*$ since
in $\2h$ we have: $\D  \hat{f} = 1/Q_2 ^2$, $\BD \hat{f} = (z-
\bar{z})^2 \f (\bar{z})/(2 Q_2 ^2) = 2 Im(z)^2 \f (\bar{z})/Q_2
^2$, $Jac(\hat{f} ) = |\D \hat{f}|^2 (1 - |\m |^2 ) \ne 0$ and
$\|\m \| < 1$.
 Therefore,
$ \hat{f}$ is a locally \q.c. \ map holomorphic in $\2h _*$. The
complex dilatation $\m$  of $ \hat{f}$ is invariant under $\G$:
recall that $|Im(\g z)| = |\g '(z) Im(z)|$, thus
$$
\m (\g z) = 2 |\g '(z)|^2 |Im(z)|^2 \f (\g \bar{z} )  =
2 \g '(z) \overline{\g '(z)} \f (\bar{z} ) \g '( \bar{z})^{-2} =
\m (z) \g '(z)/  \overline{\g '(z)}$$
and $\m (\g z)\overline{\g '(z)}/ \g '(z) = \m (z)$. So, our aim is to
show that $ \hat{f}$ is the restriction of a global \q.c. homeomorphism
of $\c$. This will be done by some approximation.

 Let $g _n$ be a sequence of Moebius
transformations such that $g _n \to id$ and $g_n (\2h _* )$ is a
relatively compact subset of $\2h _*$.
 Form the functions:
$\f _n (z) = \f (g _n z)(g _n ' (z))^2$.

Then $|\f _n (z)| = O(z^4)$ as
 $z \to \infty$ and $\f _n$ is holomorphic near $\R$.
Really,
$$
 |\f (g _n z)(g _n ' (z))^2| \le Const |g _n ' (z))^2|$$
 in $\2h _*$ since $g _n z$ belongs to a compact in $\2h _*$. Then:
$$
|g' _n (z)|^2 = |c_n z + d_n|^{-4} = O(z^{-4})$$

\begin{lem}
(Contraction property for holomorphic maps)  Let $\2h$ be a
hyperbolic plane with the Poincar\'e  metric $\rho(z) |dz|$, $\g :
\2h  \to  \2h$ is conformal. Then $d\g _z$ is ``contracting'' map
of the hyperbolic metrics: $\|\xi \| > \|d\g (\xi )\|$ for each
tangent vector at $z \in \2h$ i.e. for $w = \g (z)$ we have: $\rho
(w) |dw| \le \rho (z) |dz|$ , i.e. $\rho (\g z )|\g ' (z) | < \rho
(z)$.
\end{lem}
\proof  Assume that $\2h$ is the unit disc with
 $\rho (z) = (1- |z|^2)^{-1}$, then let $h , g \in Isom_+ (\2h )$ be such
that $g \g h (0)= 0$, $h(0) = z$. Since $g, h$ preserve the metric
its enough to prove Lemma for the point $z= 0$ and the conformal
map $f= g \g h$. However, $f(\Delta ) \subset \Delta$ and $f(0) =
0$, thus by Schwartz  Lemma: $|f' (0)| < 1$; $\rho (0) = 1$, thus
$\rho (0) |f' (0)| < \rho (0)$. \qed

The Lemma implies that
 $|g' _n (z)| < |Im(g _n (z)| /|Im(z)|$. Then
$$
|y^2 \f _n (z)| = |y^2| |\f (g _n z)| \cdot |g' _n (z)|^2 < |\f (g
_n z)|  \cdot |Im(g _n (z)|^2 \le \|\f\|$$ Thus, the hyperbolic
norm of $\f _n$ is bounded by $ \|\f\| < 1/2$.

Therefore, for the quadratic differentials $\f _n$ we can construct
the functions $\hat{f}_n$ which are  locally injective and
locally \q.c. in $\2h _* \cup \2h$ and
continuously extend to $\bar{\R}$  so that the boundary values coincide.

Moreover, $\hat{f}_n$ are holomorphic near $\bar{\R}$ and have at
worst a simple
pole at $\infty$; thus $\hat{f}_n$ is are \q.c.  homeomorphisms.

The complex dilatations $\m _n$ of  $\hat{f}_n$ have supremum
norms bounded by $\|\f \|$. On another hand, $\f _n$ are
convergent to $\f$ uniformly on compacts, thus the normalized
solutions of the Riccati equations
$$
\eta '' = -\f _n \eta /2$$
are uniformly convergent $\eta _1 , \eta _2$. Now for each $\m _n$
form the normalized solutions of the Beltrami equation. Then they are
convergent uniformly to $\hat{f}$.

 So, $\f$ belongs to the image of the Teichm\"uller space. The bicontinuity
of the correspondence $[\m ] \to S_f^{\m}$ follows from the
estimate on the norm of complex dilatation: if $\|\varphi  - \psi
\| \le \e$, then $\|\m - \nu \| \le \e$, where $\m (z) = 2 Im(z)^2
\varphi (\bar{z})$.

\qed

\medskip
The correspondence
$$
T(S) \ni [\m ] \to S_{f^{\m}}$$
is called the ``Bers embedding''.

\begin{thm}
(1) Consider the projection $p : \2h _* \to X =  \2h _* /\Gamma$.
Then the image of $Q(\Gamma )$ under $p_*$ is the space $Q(X)$ of holomorphic
\qd  s  on $X$ with at worst simple poles at the punctures of $X$.

(2) The Bers map is continuous.
\end{thm}
\proof  The elements of $Q(\Gamma )$ project to \qd  s on $X$
since they are $\Gamma$-invariant. Suppose that $\infty$ is a
parabolic fixed point of $\Gamma$ stabilized by the group $A = < z
\mapsto z + 2 \pi  >$. Then as the conformal parameter near the
puncture on $X$ corresponding to the point  $\infty$ we can choose
$w = \exp (i z)$. Denote by $D$ small \nbd of $0$ in $\c$ which is
in the image of local parameter near the puncture. Let $\f
(z)dz^2$ be $\G$-invariant, then $\f$ is invariant under $A$ and
on $D$ we have: $p_* (\f (z)dz ^2 ) = \Phi (w)dw^2 = - \Phi (w=
\exp (iz)) w^2 $. Suppose that $\Phi (w) = w^n \Psi (w)$ where
$\Psi (w)$ is holomorphic and $\Psi (0) \ne 0$. Then $\f (z) = -
\Psi (w) w^{n+2}$; $Im (z) = \log (|w|)$ and $|\log (|w|)  \Psi
(w) w^{n+2} |$ is bounded as $w \to 0$ iff  $n +2 \ge 1$, i.e. $n
\ge -1$, which means that $\Psi (w)$ has at worst a simple pole at
zero. A priori there is also case when $\Phi (w)$ has essential
singularity ar zero. However, in such case $w^k \Phi (w)$ would be
unbounded in $D$ for every $k$, thus $|\log (|w|)  \Phi (w)$ is
unbounded as well. This finishes the proof of (1).

To prove (2) its enough to show that for each $\f _n , \f _0 \in
Q(\G )$ if $\f _n \to \f _0$ uniformly on compacts in $2h _*$ ,
then $\|\f _n - \f _0 \| \to 0$ as $n \to \infty$. Let $\f _k (z)
= \Phi _k (w) w^2$, for $w = \exp (iz )$, where $\Phi _k (w) =
w^{-1} \Psi _k (w)$, $\Psi _k$ are holomorphic in $D$ and $\Psi _k
\to \Psi _0$ uniformly on compact, thus by the Maximum Principle,
$\Psi _k \to \Psi _0$ uniformly in $D$. Now,
$$
|y ^2 \f _n (z) - y ^2 \f _0 (z)| = \log ^2 (|w|) |w|^2 |w|^{-1} |\Psi _n (w)
- \Psi _0 (w)| \le $$
$$ \le |\Psi _n (w)
- \Psi _0 (w)|$$

This implies that $\|\f _n - \f _0 \| \to 0$ as $n \to \infty$.
\qed

{\bf Remark.} The property that $X$ has finite type
is essential in the proof. Each injective holomorphic function $f$ in
$\Delta$ can be uniformly on compacts approximated by functions $f_n$
with \q.c. extension to $\c$ (trick: take $f_n (z) = f(z \cdot (1- 1/n))$).
Thus, $S_{f_n}$ are convergent to $S_f$ uniformly on compacts in $\Delta$.
However, Thurston constructed examples of functions $f$ so that there
is no any sequence of injective holomorphic functions $f_k$ with the
property:
$$
\lim_{k \to \infty} \| S_{f_k} - S_f \| = 0$$

\begin{thm}
 The dimension of $Q(X)$
is equal to $3g -3 +n$ where $g$ is the genus
of $X$ and $n$ is the number of punctures.
\end{thm}
\proof  Denote by $\bar{X}$ to conformal compactification of $X$,
let $P$ be the divisor given by the set of punctures. Let $k$ be the canonical
divisor of $X$, $L(k^{-2} \cdot P )=$ set of holomorphic functions
on $X$ which have divisors at  $k^{-2} \cdot P$ at least of order
$k^{-2} \cdot P$. Then, by Riemann-Roch theorem,
$r( L(k^{-2} \cdot P )) = deg (k^2 /P ) - g +1 + r(k^2 /(Pk) = k P^{-1})$.
However, $deg (k P^{-1}) > 0$, thus $r (k P^{-1}) = 0$. On another hand,
$deg (k^2 /P ) = deg (k^2 ) - deg (P) = 2(2g- 2) - (-n) = 4g - 4 +n$;
thus $r( L(k^{-2} \cdot P )) = 3g -3 +n$. The dimension
 $r( L(k^{-2} \cdot P ))$
of $L(k^{-2} \cdot P )$ is equal to the dimension of $Q(X)$ since
$f \in L(k^{-2} \cdot P )$ iff $f \omega ^2 \in Q(X)$, where
$\omega \in \Omega (\bar{X})$ is the canonical class. \qed


\begin{thm}
 Teichm\"uller space is a manifold of the dimension $3g -3 +n$.
\end{thm}
\proof We already proved that the Bers map is a homeomorphism on
some \nbd \ $U(X)$  of $[X, id ]$ in $T(X)$. Thus, $U(X)$ is a
manifold of the dimension $3g -3 +n$. Now, let $[Y, f]$ be any
other point of $T(X)$, then we consider the homeomorphism $\alpha$
between $T(X)$ and $T(Y)$ given by : $[Z, h] \in T(Y)$ maps to
$[Z, h \circ f ]$; thus $\alpha [Y, id] = [Y, f]$. However, some
\nbd  $V$ of the point $[Y, id]$ in $T(Y)$ is also a manifold;
thus the \nbd  $ \alpha V$ of $[Y, f]$ is again a manifold.
Therefore, $T(X)$ is a manifold.  \qed

\bigskip

\begin{cor}
 The Bers' map is a homeomorphism on its image.
\end{cor}

\bigskip
\section{Poincar\'e theta series}

Let $A(\H ^2 )$ be the space of all holomorphic functions $f$ in $\H ^2$
which is realized as the unit disc in $\c$. If $\Gamma$ is a discrete torsion-
free lattice in $PSL(2, \R )$ then $A(\Gamma )$ is the space of all holomorphic functions $\varphi$ in $\H ^2$ such that :

(i) $\varphi (\gamma z ) \gamma ' (z)^2 = \varphi (z)$ for all $\gamma \in \Gamma$;

(ii) $\|\varphi \|_1 = \int _D |\varphi (z)|dx dy < \infty$ where
$D$ is a fundamental domain for the action of $\Gamma$ in $\H ^2$.

Such quadratic differentials are called "cusp forms" and their projections on $X = \H ^2 /\Gamma$ are $L^1$-integrable holomorphic quadratic differentials with at worst simple poles at the punctures. Thus, $A(\Gamma ) = Q(\Gamma )$
as linear spaces, but they are different as the normed spaces.

Now, define the operator $\Theta : A(\H ^2 ) \to A(\Gamma )$ by the formula:
$$
\Theta (f) (z) = \sum _{\gamma \in \Gamma } f(\gamma (z))\gamma ' (z)^2$$

\begin{thm}
 (1) The series $\Theta (f)$ is convergent absolutely and uniformly on
compacts in $\H ^2$;

(2) $\|\Theta \| \le 1$;

(3) The operator $\Theta$ is surjective.
\end{thm}
\proof First we recall the ``mean value'' theorem for holomorphic functions:
$$
|\varphi (w_ 0 ) = \frac{1}{2\pi r^2 } |\int _{D(w_0 , r)} \varphi |
\le \frac{1}{2\pi r^2 } \int _{D(w_0 , r)} |\varphi |
$$
for each holomorphic function in the disc $D(w_0 , r)$ with center at $w_0$ and
radius $r$.

This theorem can be proved for example via Taylor expansion for $\varphi$ with center at $w_0$.

Now, we can prove the assertion (1). Let $z_0 \in \H ^2$ and
$D(z_0 , 2r )$ be the Euclidean disc
which is contained in $\H ^2$.  Then there is a fundamental domain $D$ for the group $\Gamma$ such that $D(z_0 , 2r) \subset cl (D)$. Thus, for each $z \in
D(z_0 , r)$ we have:
$$
2\pi r^2 \sum _{\gamma \in \Gamma } |f(\gamma )| \cdot |\gamma'(z)^2 |
\le \sum _{\gamma \in \Gamma } \int _{D(z, r )} |f\circ \gamma | \cdot
|\gamma'^{2} | \le$$
$$
\le  \sum _{\gamma \in \Gamma } \int _D f\circ \gamma | \cdot
|\gamma'^{2} | = sum _{\gamma \in \Gamma } \int _{\gamma D} |f| =
\|f\|_1 < \infty $$

Now we can prove (2). Again,
$$
\|\Theta f\|_1 = \int _D |\Theta f| \le \sum _{\gamma \in \Gamma }
\int _D |f \circ \gamma | |\gamma ' |^2 $$
$$
= \sum _{\gamma \in \Gamma } \int _{\gamma D} |f| = \|f\|_{1}$$

Remark. Curt McMullen in \cite{McMullen}  proved the old
conjecture due to I.~Kra that the norm of the Theta operator is
always strictly less than 1.

We skip completely the proof of the most interesting statement (3)
since it will lead us too far from the main subject (to the theory
of Poisson kernel). You can find the proof for example in
\cite{Gardiner}. \qed

\bigskip
\section{Infinitesimal theory of the Bers map.}

 Consider the \Bd \ $\mu$ with the support in the unit disc $\H ^2$; denote by
$f= f^{t\m}$ the normal solution of the Beltrami equation:
$$
\BD f^{t\m} = t\m \D f^{t\m}$$
where $t$ is sufficiently small.
Then we recall that $f_z = h +1$ where $h = Tt\m + Tt\m (Tt\m ) + ...$:
$$
f(z) = z + \sum _{n=1}^{\infty} a_n (z) t^n$$
where
$$
a_1 (z) = P \m (z) = -\frac{1}{\pi} \int _{\H^2} \frac{z\m (\zeta )}
{\zeta (\zeta - z )} d \xi d \eta $$
Therefore,
$$
f' (z) = 1 + \sum _{n=1}^{\infty} a_n ' (z) t^n$$
$$
f'' (z) =  \sum _{n=1}^{\infty} a_n '' (z) t^n$$
$$
f''' (z) =  \sum _{n=1}^{\infty} a_n ''' (z) t^n$$
Now, our aim is to calculate the \Sd of the function $f$ in the complement
to the hyperbolic plane. First,
$$
\lim _{t\to 0} \frac{f'''}{tf'} = \lim _{t\to 0}
\frac{a_1 ''' (z) + O(t)}{1 + O(t)} = a_1 ''' (z)$$
Then,
$$
\frac{1}{t} (\frac{f''}{f'})^2 =  \frac{1}{t} (\frac{a_1 '' (t)t + ...}
{1 + ...})^2 = \frac{t^2}{t} \frac{(a_1 '' (t) + ...)^2}{(1 + ...)^2 }
= O(t)$$
Thus,
$$
\lim _{t\to 0} \frac{1}{t} S(f^{t\m }) = a_1 ''' (z) = \frac{d^3}{dz^3} (-\frac{1}{\pi} \int _{\H^2} \frac{\m(\zeta )}{\zeta - z}d\xi d \eta =$$
$$
= -\frac{6}{\pi} \int _{\H^2} \frac{\m(\zeta )}{(\zeta - z )^4}d\xi d \eta$$

This is the formula for the derivative of the Bers' map in the direction
$\m$:
$$
\dot{\Phi}(0)[\m ] (z) = -\frac{6}{\pi} \int _{\H^2} \frac{\m(\zeta )}{(\zeta - z )^4}d\xi d \eta = \sum _{n= 0}^{\infty} c_n z^{-(n+4)}\int _{\H^2} \m (z) \zeta ^n d\xi d \eta
$$
where $c_ n \ne 0$ for all $n$.
\bigskip

\begin{thm}
 Let $\Gamma \subset PSL(2, \R )$ be a torsion-free
lattice with
the fundamental domain $D$ in $\H ^2 = \Delta$.
Then \Bd \ $\m$ belongs to the kernel of $\dot{\Phi}(0)$
iff $\int _D \m \varphi = 0$ for all $\varphi \in  A(\Gamma )$. In other words,
the variation of the complex structure on $X = \H ^2 /\Gamma$ is infinitesimally
trivial along $\m \in L_{\infty} (\H ^2 , \Gamma )$ if and only if $\m$ belongs to the orthogonal complement $A(\Gamma )^{\perp}$ of
$A(\Gamma )$ in $L_{\infty} (\H ^2 , \Gamma )$.
\end{thm}

\proof  Let $\theta _n = \Theta (z^n )$; then $\int _ {\Delta} \m (\zeta ) \zeta ^n  d\xi d \eta
= \int _ {D} \m (\zeta ) \theta _n (\zeta ) $. Thus, if
$\m \in A(\Gamma )^{\perp}$ then $\int _ {D} \m (\zeta ) \theta _n (\zeta ) = 0$
and $\int _ {\Delta} \m (\zeta ) \zeta ^n  d\xi d \eta
= 0$ for all $n$, therefore, $\dot{\Phi}(0)[\m ] = 0$.

Conversely, if $\int _ {\Delta} \m (\zeta ) \zeta ^n  d\xi d \eta
= 0$ for all $n$ then we can use the Carlemann's density theorem:

polynomial functions are dense (in $L^1$ norm) in the space of $L^1$ holomorphic functions in the unit disc.

Therefore, $\m$ is orthogonal to all holomorphic functions $f$ in $\Delta$ and
$$
0 = \int _ {\Delta} \m f = \int _ {D} \m \Theta (f)$$ However, the
operator $\Theta$ is surjective, thus $\m$ is orthogonal to each
\qd  : $\m \in A(\Gamma )^{\perp}$. \qed

\begin{rem}
 This theorem is one of fundamental facts of the Teichm\"uller theory.
\end{rem}

\section{Teichm\"uller theory from the Kodaira-\newline
 Spencer point of view}

{\bf Definition} (Kodaira- Spencer): A holomorphic family is a complex manifold $V= V^{m+1}$ and a holomorphic map $\pi : V \to M= M^m$ where $M^m$ is a complex manifold and all
preimages $\pi ^{-1} (t)$ are Riemann surfaces ($t \in M$).


In our case,
$M = \Phi  (T (X)) \subset Q(X)$ and $m = 3g -3 +n$. Each point $t \in M$
corresponds to a \qd  \ $\phi _t \in Q(X)$ and to a group $\Gamma _t \subset
PSL(2, \C )$; for each $A \in \Gamma$, $A_t$ depends holomorhically on $t$.
Riemann surfaces of our family will be: $S(t) = \Omega _t  \Gamma _t$;
$$
V = \cup _{t \in M} S(t)$$
where $\Omega _t$ is a component of $\Omega (\Gamma _t )$ which is the
image of the upper half plane under quasiconformal map (thus, the variation of the complex structure on $S(t)$ isn't trivial).
Points of the space $V$ are the orbits $\Gamma _t z$ where $z \in \Omega  _t
, t \in M$. The projection is the obvious map $\pi : V \to  M$.
Now we need topology and a complex structure for the space $V$.
Consider a point $\Gamma _{t_0} z_0 \in V$. Then there is a \nbd
$N = N(z _0 )$ of the point $z_0$ such that $cl N \subset \Omega  _{t_0}$
and $\gamma _{t_0} cl N \cap cl N = \emptyset $ for all nontrivial
$\gamma $ in $\Gamma$.

Now, the \nbd \ $N(\e , z_0 , t_0 )$ of $\Gamma _{t_0} z_0 \in V$ consists of all
$\Gamma _{t} z $ such that:
$$
\|\phi _t - \phi _{t_0}\| < \e \ , \ z \in N$$ Here $\e$ is so
small number that $cl N$ doesn't meet it's $\Gamma _t - \{ 1\}$ -
orbit (such $\e$ exists since $f_t \to f_{t_0}$ uniformly on
compacts. Define the map $h$ on $N(\e , z_0 , t_0 )$ as: $h :
\Gamma _t z \mapsto (z, t)$ where $z = \Gamma _t z \cap N$.

The \nbd s $U= N(\e , z_0 , t_0 )$ define the base of topology on $V$
and the maps
$h$ are coordinate maps for the complex structure on $V$. The transition maps
are holomorphic since $A_t$ are holomorphic functions on $t$: on $U_1 , U_2$
we have:
$$
h_1 (\Gamma _t z )  = (z, t) ; h_2 (\Gamma _t z )  = (\gamma _t z, t)$$
 The projection map $\pi$ locally is given by: $(w, t) \mapsto t$ , so this
is a submersion. Therefore, $V$ is a holomorphic family.

It's much easier to visualize this construction for the case of
the Teichm\"uller space of the torus. Namely, let $\H ^2 = T(T^2
)$, $\tilde {V} = \H ^2 \times \C$. For each $t \in \H ^2$ the
lattice $\Z ^2$ acts on $\C$ as a lattice $\Gamma _t$. Now, $V $
is the quotient of  $\tilde {V}$ by this action of the group
$Z^2$.  As the result we have the fibre bundle over $\H ^2$ with
the fiber $T^2$ (which have variable complex structure). Now we
can even consider the quotient of $V$ by the modular group $PSL(2,
\Z )$. The resulting variety $U$ is fibered over the modular curve
$\H ^2 /  PSL(2, \Z )$; $U$ is called the universal elliptic
curve.

With some success we can repeat the same in the case of hyperbolic
surfaces; however instead of one and the same space $\C$ we have
to consider $\Omega _t$ as the fiber of $\tilde {V}$; $\Omega $ is
a domain in $\c$; the base of $\tilde {V}$ is the Teichm\"uller
space; $V$ is the quotient of $\tilde {V}$ by the action of $\pi
_1 (X)$ which acts as $\Gamma _t$ in each fiber. Again, we can
take the next quotient $V /Mod (X)$ to obtain the universal
Teichm\"uller curve which has the moduli space as the base and the
surface $X$ (with variable complex structure) as the fiber.

Actually, the relation between Teichm\"uller and Kodaira-Spencer
theory is much deeper.....



\section{Geometry and dynamics of quadratic differentials}

\subsection{Natural parameters}

Let $X$ be a compact \Rs , and $\f$ be a holomorphic \qd on $X$ which is different from zero. Throughout this section $\f$ is assumed to be fixed. A point $p \in X$ is said to be {\bf regular} with respect to $\f$ if $\f (p) \ne 0$ and {\bf critical} if $\f (p) =0$. It's easy to see that these definitions do not depend on the choice of local coordinates on $X$.
Critical point of $\f$ form a finite set $C(\f )$. Let $p$ be any regular point and
$q \mapsto h(q) = z$ is a local coordinate near $p$ such that $h(p) =0$.
Since $\f (p) \ne 0$ then there is a small \nbd of $0$ where who branches of $\sqrt {\f (z)}$ are single valued. For a fixed branch of square root every integral
\BE
z \mapsto \Phi (z) = \int_{0}^{z}  \sqrt {\f (w)}dw
\EE
is also a single-valued function in some simply-connected \nbd of
 $0$ and uniquely determined up to an additive constant.

On another hand,
$\Phi ' (0) = \sqrt {\f (0)} \ne 0$ and thus, $\Phi$ is locally injective near
$0$. It follows that the system of maps $z \mapsto w= \Phi (z= h(q))$
is a holomorphic atlas on $X- C(\f )$. In these local coordinates
$\f (z)dz^2 = dw ^2$. The coordinate $\Phi$ is called a {\bf natural parameter}
near $p$. An arbitrary natural parameter near $p$ has the form $ \pm z + const$. This means that the natural parameters define a very special kind if Euclidean structure on $X - C(\f )$ (which is called $F$-structure, where $F$ stands for the ``foliation'').

There are natural parameters at the critical points as well.  Suppose that
$p \in X$  is a zero of order $n$ for $\f$. Again, let $q \mapsto h(q)= z$
be a local parameter near $p$. Then there is a disc $D= D(0, r)$ where
$\f (z) = z^n \psi (z)$ with $\psi (z) \ne 0$. We fix a single-valued branch of $\sqrt {\psi}$ in  $D$. If $n$ is odd then we cut $D$ along $\R _+$ and fix a
branch of $z \mapsto z^{n/2}$ in $D' = D - \R _+$
; if $n$ is even we don't need any cut. In any case,
\BE
z \mapsto \Phi (z) = \int_{0}^{z} \sqrt {\f (w)}dw = z^{(n+2)/2}(c_0 + c_1 z + ...) = z^{(n+2)/2} \omega (z)
\EE
(where $c_0 \ne 0$) is a single-valued function in $D'$.
Moreover, the function
\BE
z \mapsto \omega(z) = \Phi (z) z^{-(n+2)/2}
\EE
is single-valued at some \nbd of $0$. This,
$$
\zeta: z \mapsto \Phi (z)^{2/(n+2)} = z \omega (z)^{2/(n+2)} \eqno(4)$$
is single-valued near $0$ since $\omega (z) \ne 0$;
and has nonzero derivative at $0$:
$$
\zeta ' (0) = \omega (0)$$
We call
$\zeta: q \mapsto \Phi (z)^{2/(n+2)}$ to be the {\em natural parameter} at $p$. In terms of this natural parameter we have:
$$
\f dz^2 = (\frac{n+2}{2})^2 \zeta ^n d\zeta ^2 \eqno(5)$$
since:
$$
d\zeta = \frac{2}{n+2} \Phi ^{\frac{2}{n+2} -1 }(z \Phi ' (z) dz =
\frac{2}{n+2} \Phi ^{\frac{-n }{n+2}}\sqrt{\phi (z)}dz$$
$$
\f dz^2 = (\frac{n+2}{2})^2 \Phi ^{\frac{2n }{n+2}}d\zeta ^2 =
(\frac{n+2}{2})^2 \zeta ^n d\zeta ^2$$

Define the differential $|\f (z)|^{1/2}|dz|$. This is a Riemannian metric outside the critical set of $\f$. This metric is locally Euclidean on
$X - C(\f )$ and is called $\f $-metric. The natural parameter is the local isometry between this metric and the Euclidean metric on $\C$. The
surface $X$ has a finite diameter and area with respect to this singular metric. We shall return to this metric later.



\subsection{Local structure of trajectories of \qd s}

Horizontal (or vertical) trajectories of $\f$ correspond to the (maximal) horizontal
(or vertical) Euclidean line in $\c$ under the natural parameter
(on $X - C(\f )$). Another way to define these trajectories is as follows.
Let $\gamma : [-1 , 1] \to X - C(\f )$ be a smooth path, take the pull-back
$$
\f (\gamma (t)) (\gamma ' (t))^2 dt^2 \eqno(6)$$
of the form $dz^2$ on $[-1 , 1]$. Then the curve $\gamma$ is called a straight
line if the argument $arg(\f (\gamma (t)) (\gamma ' (t))^2 ) = \theta$ is constant. The trajectory is called horizontal if $\theta = 0$ and vertical
if $\theta = \pi $.

Near singular points the trajectories are more complicated. Let $\zeta = w^{2/(n+2)}$ be the
natural parameter near  a critical point $p$; then $w$ is a natural parameter
outside of $p$.

If $p$ is zero of order $n$ for $\f$ then there are $n+2$ horizontal rays emanating from $p$; in the natural parametrization the angles between them are $2\pi /(n+2)$. See Figure 5.

 \begin{figure}[tbh]
\centerline{\epsfxsize=3in \epsfbox{fig5.eps}}
%\caption{m-fold watermelon covering}
%\label{fig:cyclic m-fold covering}
\end{figure}

Trajectory is called {\em critical} if it contains one of critical points.
From now on we shall consider only horizontal trajectories of $\f$.

{\bf Examples on the torus.} Suppose that $X$ is a torus obtained by identification of sides of a parallelogram $P \subset \C$; denote by $\f$
the projection on $X$ of the \qd ~ $dz^2$. In this case $C(\f ) = \emptyset$. Then the natural parameter on $X$
is the inverse to the universal covering and restriction of it to $P$ is the identity map. The horizontal trajectories of $\f$ are projections on $X$ of the horizontal lines in $\C$.
Suppose that $P$ is a rectangle. Then all horizontal trajectories of $\f$ are closed parallel geodesics on $X$. However, in the generic case, the trajectories of $\f$ are irrational lines which are dense on $X$.

Now we consider the case of surface of general type. To simplify the discussion we shall assume that $X$ is compact.
First notice that  $C(\f ) \ne \emptyset$ since $\chi (X) \ne 0$. We have the following classes of trajectories:

(a) Periodic trajectories. Let $\g$ be a periodic trajectory of a \qd . We shall see that  there is a maximal open annulus $A$ on $X$ which contains $\g$ and which is foliated by closed trajectories of $\f$ and which has no critical points.

(b) Critical trajectories. These are trajectories $\gamma$ such that at least one ray of $\gamma$ end in a critical point of $\f$. There is only a finite number of such trajectories.


(c) Nonperiodic noncritical ({\bf spiral}) trajectories $\gamma$. We shall see that they  are recurrent  in positive and negative direction. This means that $\gamma$ is contained in the limit set for both rays
$\gamma _+$ and $\gamma _{-}$.

Now, let's discuss the trajectories in more details.

\bigskip
\subsection{Dynamics of trajectories of \qd }

Suppose that $\g$ is a (horizontal) trajectory of $\f$. Let $p \in \g$,
$\Phi (p) = 0$, $\Phi (\g ) \subset \R \subset \C$. Let $I= [a, b]$ be the maximal open interval on $\R$ which contains $0$ such that the inverse to $\Phi$ is defined there. Denote the inverse by $f$.

(a) First suppose that $I$ is bounded and $\Phi ^{-1} (a) = \Phi ^{-1} (b)$.
This implies that $\g$ is a periodic trajectory of $\f$. In this case we can consider  the maximal horizontal strip $]a, b[ \times ]x, y[ \subset \C$
where $f$ is defined and injective. The image of $\R \times ]x, y[$ is an
annulus $A$ in $X$ foliated by trajectories of $\f$. If there are points on
$I \times \{x\}$ and $I \times \{y\}$ which have the same image under $f$ then
$X$ is a torus. So we can assume that $f$ is an injective holomorphic function
on $I \times [x, y]$. However, we can't extend $f$ through  $I \times \{x\}$ and $I \times \{y\}$ which implies that both sides of $A$ contain critical trajectories. Example of this type of behavior on a pair of pants is shown on the Figure 6. The maximal annulus $A$ has the geometric invariant- height (i.e.
$|x- y|$).
There is a class of differentials which have only periodic and
critical trajectories (of finite length). These differentials are
 called {\bf Strebel differentials}.
Moreover, given a maximal collection
of pairwise disjoint nonhomotopic loops $\gamma _j$ on $X$ and a collection of positive numbers $h_j$ there is a unique Strebel \qd ~  $\f$ on $X$ such that
the maximal annuli $A_j$ of $\f$ are homotopic to $\gamma _j$ and have the prescribed heights $h_j$ (see \cite{Gardiner}). Strebel differentials are dense in $Q(X)$.

 \begin{figure}[tbh]
\centerline{\epsfxsize=3in \epsfbox{fig6.eps}}
\caption{}
%\label{fig:cyclic m-fold covering}
\end{figure}



(b) Consider the case when $I \ne \R$ and the points on the boundary of $I$
have different image under $f$. Then $\gamma$ is a critical trajectory and
if $z$ is a boundary point of $I$ then $w=f(z)$ is a critical point of $\f$.
If $z$ is the right end of $I$ then the positive ray of $\gamma$ has unique
 limit point $w$.

\medskip
(c) The most interesting case is when (say) the positive ray $\g _+$ of
 $\gamma$  has more than one limit point and $\gamma$ isn't periodic. Then
$\g _+$ has necessarily infinite (Euclidean) length. Really, if $q \in
L(\g _+ )$ then $\g _+$ intersects any  \nbd of $q$ infinitely many times ,
thus the length of intersection of some \nbd of $q$ is bounded away from zero
and $\g _+$ has infinite length. Suppose that $\gamma$ isn't critical; then
both rays $\g _+$ and $\g _{-}$ have infinite length and  the map $f$ is
defined and injective on the whole line $\R$.

\begin{thm}
 If $\g$ is the trajectory of the type (c) which is infinite
 in the positive direction then the ray $\g _+$ is recurrent.
\end{thm}
\proof Let $p \in \g$ and $\beta$ is a vertical interval with
endpoint in $p$. We can assume that $\beta$ is so small that no
critical (positive) ray intersects this interval. Suppose that for
all positive rays $\alpha$ emanating from points of $\beta$ ,
$\alpha \cap \beta $ is the origin of $\alpha$. Then the infinite
horizontal strip $S$ with base at $\beta$ is embedded in $X$
(since trajectories form a foliation on $X - C(\f )$. However, $S$
has infinite area. This contradicts the finiteness  of the area of
$X$ with respect to the metric $|\f (z)|^{1/2}|dz|$. \qed

\begin{cor}
 The  intersection $J$ of $\gamma _+$ with a small
interval $\beta$ above is dense in $\beta$.
\end{cor}
\proof The closure of the intersection above is a perfect subset.
On another hand, let $z$ be a boundary point of $Cl(J )$ in
$\beta$. Denote by $\sigma$ a positive ray emanating from $z$. We
can assume that $\sigma$ isn't critical, thus, it intersect
infinitely many times each small interval adjacent to $z$ on
$\beta$. But $\sigma$ is contained in the closure of $\gamma _+$,
thus the points of $J$ accumulate to $z$ from 2 sides on $\beta$.
This contradicts to the assumption that $z$ is a boundary point.
Thus, $cl(J) \supset \beta$. \qed

\medskip
Now, consider the closure of $\gamma$. The arguments of Theorem above imply that $A= Cl(\gamma )$ has nonempty interior $A^0$ which is called a maximal spiral domain in $X$. As in the case of periodic trajectories this domain is swept by parallel trajectories.

\begin{lem}
 The boundary of $A$ consists of finite critical trajectories
 of $\f$ and their limiting points.
\end{lem}
\proof Let $\alpha \subset \D A$ is infinite in the positive
direction and $P \in \alpha$ be a regular point. Then for some
small vertical interval $\beta$ with center at $P$  the ray
$\alpha _+$ intersects $\beta$ on dense subset (see Corollary
above). Thus $A$ contains a \nbd of $P$ which contradicts to our
assumption.  \qed

\subsection{Examples of spiral trajectories.}

As you can see, it's rather difficult to construct examples of spiral
 trajectories. To do this we shall use the construction of ``interval
 exchange'' transformations. Take the  horizontal rectangle $S$ in the
 complex plane :
$\{ z : 0 \le Im (z) \le 1 , -1 \le Re (z) \le 1 \}$. On the segment
$\{ 0\} \times [0, 1]$ we choose the intervals: $I_1 ^+ = [0, x]$,
$I_2 ^+ = ]x , y] ,  I_3 ^+ = ]y , 1]$ and:
$$
I_1 ^- = [0, 1-y] , I_2 ^- = ]1-y , 1-x ] , I_3   = ]1-x , 1]$$
Now, to each pair of intervals $I_k ^+ , I_k ^-$ we glue a rectangle
$S_k$ of the width 1 in orientation preserving way. We identify the "adjacent"
rectangles by 1/3 of their width. In the bifurcation points we introduce the
 local complex coordinates using the square root. The result is a \Rs
~$Y$ with boundary, then we can take the double $X$ of $Y$ by reflection.
The surface $X$ has the \qd ~$\f$ which is just the projection of $dz^2$
from the complex plane. The critical points are the points of bifurcation.
Generically the trajectories of $\f$ are recurrent or critical. (Figure 7)


 \begin{figure}[tbh]
%\leavevmode
\centerline{\epsfxsize=3in \epsfbox{fig7.eps}}
\caption{}
%\label{fig:cyclic m-fold covering}
\end{figure}



\subsection{Singular metric induced by \qd }

Define the differential $|\f (z)|^{1/2}|dz|$. This is a Riemannian metric
outside the critical set of $\f$. This metric is locally Euclidean on
$X - C(\f )$ and is called $\f $-metric. The $\f$-length of curve is
 also defined in the case when the curve is passing through a critical point.
the total angle around the critical point of order $n >0$ is equal to $(n+2)\pi > 2\pi$. This metric is so called $CAT(0)$.

Consider the lift of the $\f$-metric to the universal cover of $X = \H ^2$,
suppose that $P$ is a polygon in $\H ^2$ with geodesic sides $E_j$. Denote by
$\tilde{\f}$ the lift of $\f$. Let $z_j$ denote a zero of $\tilde{\f}$
on $P$ where the interior angle (in the hyperbolic metric) between adjacent edges is $\theta _j$, $n_j$ denote the order of $z_j$ (it can be zero).  Denote
by $w_i$ zeros inside  $P$ with orders $m_i$. Let $m$ be the number of zeros inside of $P$. Then we have

\begin{lem}
\label{TGB} (Teichm\"uller-Gauss-Bonnet). In the notations above
we have:
$$
\sum _j (1- \frac{\theta _j (n_j +2)}{2\pi }) = 2 + \sum _i m_i = m+2 \eqno(7)
$$
(This is a combinatorial Gauss-Bonnet formula for the $\f$-metric).
\end{lem}
\proof  Along the sides $E_j$ we have: $arg(\f dz^2) = const_j$.
Let $z= h_k (t)$ be a parametrization of $E_k$. Then
$$
arg(\f dz^2 ) = arg [\f (h_k (t)) (h_k ' (t))^2 ] =
arg [\f (h_k (t))] + 2arg[ h_k ' (t) ]$$
 therefore,
$$
\frac{d}{dt}\arg \f (z(t)) = -2  \frac{d}{dt} arg [ h_k ' (t) ]$$
abusing notations we can write this as:
$$
\frac{d}{dz}\arg \f (z) = -2  \frac{d}{dz} arg [dz ]$$
along $P$. However the increment of $arg [dz ]$
along $P$ is equal to
$$
2\pi - \sum _j (\pi - \theta _j )$$
(if curve would be smooth then the total increment is $2\pi$, in each corner
we arrive with angle smaller by $\pi - \theta _j$ than the expected vector- tangent to the next arc).

Thus we have:
$$
\frac{1}{2\pi } \int _P d arg[\f (z) ] = -\frac{2}{2\pi } \int _P d
arg[dz  ]$$
The last integral is equal to:
$$
-2 + \sum_j (1- \frac{\theta _j }{\pi })$$
According to the argument principle the first integral is:
$$
\sum_i m_i + \sum_j \frac{n_j\theta _j}{2\pi}$$
Therefore,
$$
\sum_i m_i + \sum_j \frac{n_j\theta _j}{2\pi} =
 -2 + \sum_j (1- \frac{\theta _j}{\pi })$$
This implies the lemma. \qed

\begin{cor}
 The $\f$-geodesic between two points of $\H ^2$ is unique.
\end{cor}
\proof  Suppose that we have a geodesic bigon $P$ in the
$\f$-metric. The Lemma \ref{TGB} implies that at least 3 summands
in the left side of (7) are positive since the left side is $\ge
2$. But this means that we  have at least 3 points with $\theta _j
< \frac{n_j +2 }{2\pi}$. Thus, we have at least one point  on the
side of bigon such that the angle at this point is less than
$\frac{n_j +2 }{2\pi}$. Therefore,this side of the bigon isn't
geodesic. \qed

\medskip
Now, let's count the sum of angles in a geodesic triangle in our metric.
We have:
$$
\sum_{j= 4}^{k} (2\pi - \theta _j (n_j + 2)) + 6 \pi - (\theta _1 + \theta _2
+ \theta _3 ) = 2\pi (n+2) \eqno(8)$$
The summands $(2\pi - \theta _j (n_j + 2)) = \epsilon _j$ are nonpositive
since we have only 3 vertices, thus:
$$
\theta _1 + \theta _2
+ \theta _3 = 2\pi (n -1) + \epsilon \eqno(9)$$
where $ \epsilon \le 0$. Therefore, $m = 0$, sum of angles in the triangle
is   $\le 2\pi$ and the triangle can't contain any critical points inside.


Thus, the radius of the inscribed  disc in any geodesic triangle is bounded
 from above by the diameter
of $(X , \f )$. (It's impossible to find a bound independent  on $\f$.)
%Therefore, we have a hyperbolic space in the sense of Gromov.

\subsection{Deformations of horizontal arcs. }


\begin{lem}
%{\bf Lemma 3}
(Teichm\"uller). Let $X$ be a compact \Rs , $f: X \to X$ be a
homeomorphism homotopic to identity and $\a$ is a horizontal arc.
Then there exists a constant $M$ independent on $\a$ such that:
$$
l(f (\a )) \ge l(\a ) - 2M \eqno(10)$$
where $l(.)$ is the $\phi$-length.
\end{lem}
\proof Let $f_t$ be the family of continuous maps such that $f_0 = id$,
$f_1 = f$. For each point $p \in X$ consider the displacement function:
$d_f (p) = l_{\f} (\gamma _p )$
where $\gamma _p$ is the $\f$-geodesic connecting $p$ and $f(p)$ which belongs to the homotopy class of the path $f_t (p) , t\in [0, 1]$. The function $d_f (p)$  is continuous, denote its maximum on $X$ by $M$. Now, connect the
endpoints $p, q$ of the horizontal arc to $f(p) , f(q)$ by the geodesic segments $\gamma _p$, $\gamma _q$. We have : $l(\g _p ) , l(\g _q ) \le M$,
$$
length ( (\g _q )^{-1} \cdot f(\a ) \cdot (\g _p )) \le l(\a )$$
thus $l( f(\a )) + 2M \ge l(\a )$. \qed

\begin{cor}
 Under conditions  above:
$$
\lim _{l(a) \to \infty} l(f(\a ) )/l(\a ) \ge 1$$
\end{cor}

\subsection{Orientation of the horizontal foliation}

Suppose that the monodromy group of the natural parameter $\Phi$
consist only of translations. Then the horizontal foliation of $X
- C(\f )$ admits an orientation which is just the pull-back of the
orientation of horizontal lines in $\C$. In general however there
exists a nontrivial character $\rho : \pi _1 (X - C(\f )) \to
U(1)$ given by the linear part of the monodromy of $\Phi$. Then
$Ker (\rho )$ is a subgroup of index 1 or 2 in $\pi _1 (X - C(\f
))$. The 2-fold ramified covering $q: X_0 \to X$ corresponding to
this subgroup  is called the orienting covering of $X$. The
pull-back $q^* (\f ) = \psi$ is a \qd ~ on $X_0$ which has only
even zeros and whose horizontal foliation admits a global
orientation.




\section{Extremal quasiconformal mappings}
\bigskip

\subsection{Extremal maps of rectangles}

We shall use the proof of the following theorem as the model for proof of
 the extremality theorem in the general case.

Recall that
$$
K_f (z) = \frac{|\D f| + |\bar{\D}f|}{|\D f| - |\bar{\D}f|}$$
and the coefficient of quasiconformality of $f$ is
$$
K_f = esssup_z K_f (z)$$

\begin{thm}
 (Gr\"otch). Let $R, R'$ be  rectangles: $a\times b$ and
$a' \times b'$. Suppose that $f: R \to R'$ be a diffeomorphism.
Then $K_f \ge K_0 = (a'/a) \cdot (b/b')$ and equality is achieved only
on affine maps.
\end{thm}
\proof Put $z= x + iy$.
Let $\a$ be a horizontal line in $R$, then
$$
a' \le \int_{\a} |f_x| dx \eqno(11)$$
Taking integral over $y$ we have:
$$
a'b \le \int_{R} |f_x |dx dy \eqno(12)$$
However, $f_x = f_z + f_{\bar{z}}$ and Jacobian of $f$ is
$$
J_f (z) = |f_z |^2 - |f_{\bar{z}}|^2 \eqno(13)$$
Therefore,
$$
(|\D f | + |\bar{\D}f|)^2 = K_f (z) J_f (z)$$
Then
$$
a'b \le \int_{R} |f_z + f_{\bar{z}}|dx dy \le  \int_{R} \sqrt{K_f (z)}
\sqrt{J_f (z) } dx dy \eqno(14)$$
The last integral is estimated as
$$
\sqrt{\int_{R} K_f (z)} \sqrt{\int_{R} J_f (z)} \le \sqrt{K_f } \sqrt{ab}
\sqrt{a'b'} \eqno(15)$$
Finally we have:
$$
(a'b)^2 \le K_f ab a'b' $$
i.e.
$$
a'b \le K_f ab'$$
and we are done. The equality here is achieved only under conditions:

(a) $K_f = (a'/b')/(a/b) = K_0$

(b) $Im (f) _x = 0 , Re(f)_y =0 $

(c) $J_f = Re(f)_x Im(f)_y = c \cdot K_f = c Re(f)_x / Im(f)_y$.

Thus, $f$ is a linear function. \qed

\section{Teichm\"uller differentials}

Let $\f \in Q(X)$  be a nonzero \qd . Then for each $0 \le k < 1$ we
define
$$
\mu = k\frac{\bar{\f}}{|\f |} \eqno(16)$$
It's easy to see that $\mu$ is a \Bd \ on   $X$. Then the \Rs ~ $Y$
with complex structure determined by $\mu$ is quasiconformally  equivalent to
$X$. The new \Rs has natural marking and thus defines a point in $T(X)$. This
 deformation of the original complex structure on $X$ is called
{\it Teichm\"uller deformation}.

Let's look at this deformation in terms of the natural parameter near regular
 points.
Suppose that $z$ is the natural parameter , then $\f = dz^2$ and $\m (z) = k$.
Solutions of the Beltrami equation with the characteristic $\m$ which fix the
 points $\infty$ are affine maps. The push-forward of $\f$ under $f^{\m}$
is $\psi \in Q(Y)$. In terms of the natural parameters $z , \zeta$
 corresponding to $\f$ and $\psi$ we have:
$$
\psi = (d\zeta )^2 , f(x + iy) = K x +iy = \zeta$$
 where $K = (1+k)/(1-k)$. The affine maps $f$ as above form a new
 $F$-structure on $Y$ since the group $\R ^2 \times O(1)$ is normal in
$Aff (\R ^2 )$ and therefore, the transition maps are as above.
Therefore, the map $f$ is an affine horizontal stretch in terms of
the natural parameters. For the Teichm\"uller mapping $f$ the \qd
s $\f , \psi$ are called {\it initial } and {\it terminal}
respectively. In such case $(Y , \psi ) = f(X, \f , k )$.

\section{Stretching function and Jacobian}

Suppose that we have $(X, \f )$ and $(Y, \psi ) = f(X, \f , k )$. Suppose that
$g : X \to Y$ be any \q.c. ~ homeomorphism. Then, if $z= x + iy$  is the natural parameter on $X$, and $w = g(z)$ be the natural parameter on $Y$, then
$$
\l _{g, \f , \psi} (z) = \l _g (z) = |w_x | \eqno(17)$$
In terms of the conformal structures on $X, Y$ we have:
$$
\l _g (p) = |\frac{\D g(p)}{\f (p) ^{1/2}} + \frac{\bar{\D g(p)}}{\bar{\f }(p) ^{1/2}} |\cdot |\psi (g(p)|^{1/2} \eqno(18)$$
Then $\l _g (p)$ is a function on $X$. Let $\a$ be any horizontal arc on $X$.
Then for all but finitely many of trajectories we have:
$$
\int_{\a} \l _g |\f |^{1/2} = length (g(\a )) \eqno(19)$$
The Jacobian of $g$ in terms of the natural parameter is:
$$
J_g (z) = |\D w |^2 - |\bar{\D }w|^2 \eqno(20)$$
this is the same as
$$
J_g (z) = (|\D w |^2 - |\bar{\D }w|^2 ) \frac{|\psi (w (z))| }{|\f (z)|}
\eqno(21)$$
which is a function on $X$. Therefore,
$$
Area_{\psi} (Y) = \int_{X} J_g |\f | \eqno(22)$$

\section{Average stretching}

\begin{thm}
\label{ave}
Let $g: X \to X$ be a \q.c. homeomorphism homotopic to
 identity and
$\f \in Q(X)$. Define $\l _g$ for $\f = \psi$. Then:
$$
\int_X \l _g |\f | \ge Area_{\f}(X) \eqno(23)$$
\end{thm}
\proof First we define a 1-dimensional average. Let $\a$ be a subarc of a
 horizontal trajectory with midpoint $p$ and of length $2a$. We set:
$$
\l _a (p) = \frac{1}{2a} \int_{\a}\l |\f |^{1/2} \eqno(24)
$$
Assume for a moment that $(X, \f )$ has oriented trajectory structure. Let
$X_0$ be the union of noncritical trajectories. This set has full measure on
$X$. We define a flow on $X_0$. Let $p$ be a point on a horizontal trajectory
$\a \subset X_0$, $t \in \R$. Let $\chi (p, t)$ be the horizontal translation
of $p$ to the time $t$. Then $\chi (., t)$ is isometry for each $t$. It
 follows that $\l ^t = \l \circ \chi (., t)$ is a measurable function
on $X_0$ and
$$
\int_X \l ^t |\f | = \int_{X_0} \l ^t |\f | = \int_X \l  |\f | \eqno(25)$$
Hence,
$$
\int_X \l  |\f | = \frac{1}{2a} \int_{-a}^{a} (\int_{X} \l ^t |\f | )dt =
\int_{X} (\frac{1}{2a} \int_{-a}^{a} \l ^t  dt )|\f | = \int_{X} \l _a |\f |
$$
Therefore,
$$
\int_X \l  |\f | = \int_{X} \l _a |\f | \eqno(27)
$$
If the trajectory system isn't orientable, then we pass to the
2-fold branched covering of $X$, where all integrals double, thus
the identity (27) is valid in this case as well. Then, according
to Teichm\"uller's lemma,
$$
\l _a (p) \ge 1 - M/a \eqno(28)$$
almost everywhere and
$$
\int_X \l  |\f | \ge(1- M/a)Area _{\f} (X) \eqno(29)$$
Finally, letting $a \to \infty$ we obtain
$$
\int_X \l  |\f | \ge Area _{\f} (X) \eqno(30)$$ \qed

\begin{cor}
\label{Corollary 5}
 If $g$ is as above and $\|\psi \|_{L_1} \ge 1$ then
$$
Area_{\psi} (Y)  \le \int_{Y} \l ^2 _{g, \psi , \psi} dA_{\psi} \eqno(31)$$
\end{cor}
\proof According to Theorem \ref{ave} and Schwarz inequality we have:
$$
Area_{\psi} (Y)  \le \int_{Y} \l _{g, \psi , \psi} dA_{\psi} \le
\int_{Y} \l ^2 _{g, \psi , \psi} dA_{\psi} $$ \qed

\subsection{Teichm\"uller's uniqueness theorem}

\begin{thm}
 (Teichm\"uller's uniqueness theorem).
Let
$$
f: X \to (Y, \psi ) = f (X, \f , k)$$
 be a homeomorphism of $X$
homotopic to identity. Then
$$
K(f) \ge K_0 = (1+k)/(1-k) \eqno(32 )$$
The equality takes place only if $f = id$.
\end{thm}
\proof  Without loss of generality we can assume that $Area_{\psi}
(Y) = \|\psi \|_{L_1} = 1$. Denote by $id$ the identity map of
$X$. First we notice that:
$$
\l _{f, \f , \psi} = K_0 \l _{f, \psi , \psi} \eqno(33)  $$
$$
J_{id , \f , \psi} = K_0 ~~,~~ dA_{\psi} = K_0 dA_{\f} \eqno(34)$$

\begin{claim}
$$
\l ^2 _{f, \f , \psi} \le K(f) J_{f, \f , \psi} \eqno(35)$$
\end{claim}
\proof Let the natural parameter on $X$ be $z = x + iy$, and on $Y$:
$w= u + iv = f(z)$. Then
in the local coordinates we have:
$$
K_f (z) J_f (z) =  \frac{|f_z | + |f_{\bar{z}}|}{ |f_z | -
|f_{\bar{z}}|} |f_z |^2 - |f_{\bar{z}}|^2 = (|f_z | +
|f_{\bar{z}}|)^2  \ge (|f_z + f_{\bar{z}}|)^2 = \l ^2 _{f, \f ,
\psi} \eqno(36)$$ This proves the claim. \qed

Now, applying (33- 35) we get:
$$
\int_Y \l _{f , \psi , \psi} ^2 dA_{\psi} =
K_0 ^{-2} \int_Y  \l _{f , \f , \psi} ^2 dA_{\psi} =
K_0 ^{-1} \int_X  \l _{f , \f , \psi} ^2 dA_{\f} \le$$
$$
\frac{K[f]}{K_0 } \int_X J_{f, \f , \psi} dA_{\f} = \frac{K[f]}{K_0 }
\int_Y dA_{\psi} = \frac{K[f]}{K_0 } Area_{\psi }(Y) \eqno(37)$$
However, according to Corollary \ref{Corollary 5},
$$
Area_{\psi} (Y)  \le \int_{Y} \l ^2 _{g, \psi , \psi} dA_{\psi}$$
Thus, $K[f] \ge K_0$.  Therefore, we proved (32).

Suppose now that in (32) we have the equality, in particular,
$$
|\bar{\D} w + \D w|^2 = \l _{f , \f , \psi} ^2 = K_0 J_{f, \f , \psi} =
K_0 (- |\bar{\D} w |^2  + |\D w|^2 )
 \eqno(38)$$
$$
|\bar{\D} w| = k_0 |\D w| \eqno(39)$$
Denote $\D w $ by $r(z) e^{i\theta (z) }$, then $\bar{\D} w = k_0 r(z) e^{i\nu (z)}$
and (38) can be written as:
$$
|e^{i\theta} + k_0 e^{i\nu}|^2 = K_0 (1 - k_0 ^2) = (1 + k_0 )^2 \eqno(40)$$
i.e.
$$
|1 + k_0 e^{i(\nu - \theta )}| = 1 + k_0$$
which is possible only if $\nu = \theta$. This means that
$$
\bar{\D} w  = k_0 \D w$$ and $f$ is a conformal mapping from $Y$
to $Y$. Therefore, $f = id$. \qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{cor}
\label{Corollary 6}
 If $Y = f(X, \f , k) = f(X, \psi , t)$ then $\f /\psi \in \R$,
$t = k$.
\end{cor}

\subsection{Teichm\"uller's existence theorem}


Denote by $T_e (X)$ the space of pairs $(\f , k)$, where $\f \in Q(X)$
has norm $1$, $k \in (0, 1)$, and the pair $(0, 0)$. This space has natural topology given by supremum norm on $Q(X)$ and is homeomorphic to the open unit ball of dimension $3g -3$ in $Q(X)$. Denote by
$F$ the natural map from $T_e (X)$ to $T(X)$:
$$
F : (\f , k) \mapsto f^{\mu} , \mu = k\bar{\f}/|\f |$$

\begin{thm}
%{\bf Theorem 5.}
The map $F$ is a homeomorphism on the Teichm\"uller space.
\end{thm}
\proof Then this map is injective according to Corollary \ref{Corollary 6}. This map is continuous since solution of Beltrami equation depends continuously on the characteristic. We have:
$$
\lim_{n \to \infty} d_T [F(0)= X , F(\f , k_n = (1-1/n))] =
\lim_{n \to \infty} \log \frac{1+ k_n}{1- k_n} = \infty$$
and thus for any bounded subset $C \subset T(X)$ its preimage
$F^{-1} (C)$ is relatively compact in $T_e$. On another hand, the map
$F$ is open since the spaces $T(X)$ and $Q(X)$ are manifolds of the same dimension $3g -3$.
Therefore, $F$ is a surjection on the connected component of $T(X)$.
But as we know, $T(X)$ is connected, thus $F$ is a surjection. Therefore
 $F$ is a homeomorphism. \qed

\subsection{Teichm\"uller geodesics}

We recall that $\Delta = \{ z \in \C : |z| < 1\}$ is the unit disc model of
the hyperbolic plane.

\begin{thm}
\label{Theorem 6}
 For each $\f \in Q(X) - \{ 0\}$ the map
$$
h_{\f} : \Delta \to T(X)$$
defined by the formula
$$
h_{\f} : t \mapsto [f^{\mu} ]~~,~~ \mu = t\bar{\f }/|\f |$$ is an
isometry of the hyperbolic plane into the Teichm\"uller space.
\end{thm}
\proof We know that
$$
d_{T} (0 , [f^{\mu} ]) = \log \frac{1 +|t|}{1 - |t|}$$ since
Teichm\"uller maps are extremal. On another hand, the hyperbolic
distance between $0$ and $t$ in $\Delta$ is equal to
$$
\log \frac{1 +|t|}{1 - |t|}$$
Thus, the map $h_{\f}$ preserves the distance from $0$ to $t$. To prove the assertion in general case we need

\begin{lem}
%{\bf Lemma 4}.
Suppose that $\psi = t \f \in Q(X) - \{0\}$. Then the composition
$$
f^{k\bar{\psi}/|\psi |} \circ (f^{r\bar{\phi}/|\phi |} )^{-1}$$ is
again a Teichm\"uller map.
\end{lem}
\proof First we notice that if $f: (X, \phi , k) \to (Y, \psi )$
is a Teichm\"uller map then $f^{-1}$ is the Teichm\"uller map $(Y,
- \psi , k) \to (X, -K^2 \f )$. Really, ...

Now, let's prove the assertion of Lemma. Denote
$f^{k\bar{\psi}/|\psi |}$ by $f_2$ and $f^{r\bar{\phi}/|\phi |}$
by $f_1$. Denote by $\zeta , \zeta ^*$ the natural parameters for
$\f$, $\psi$ and $\zeta _1$, $\zeta _2$ the natural parameters for
the terminal differentials of  $f_1 , f_2$. Consider the map $f_2
\circ (f_1 )^{-1}$ in terms of $\zeta _1$, $\zeta _2$. It can be
presented as composition $A \circ C \circ B$ where $B : \z _1
\mapsto \z$, $C: \z \mapsto \z ^*$, $A: \z ^* \mapsto \z _2$. Here
$A, B$ are stretchings and $C$ is a conformal map $\z \mapsto \z
\cdot \sqrt{\psi /\phi } = \z a e^{-i\theta /2}$, where $a > 0 , 0
\le \theta < 2\pi$. \qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

\section{Discreteness of the modular group}

Our aim is to prove the following

\begin{thm}
\label{Theorem 7}
 The action of $Mod_X$ on $T(X)$ is properly discontinuous.
\end{thm}

We already know that this action is isometric. Thus, it's enough to prove that any orbit of $Mod_X$ has no accumulation points.

Now, let $[X]$ be the origin of $T(X)$, $[Y, f] \in T(X)$. Then the ``length
spectrum'' of $[Y, f]$ is the function
$$
{\cal L} : \g \mapsto length_{Y} [f(\g )]$$
where $[f(\g )]$ is the geodesic in the homotopy class of $f(\g )$,
$\g \in \pi _1 (X) /Inn (X)$. The image of the function ${\cal L}$
is called the {\bf length spectrum} of $Y$ and denoted by $L(Y)$.

\begin{lem}
%{\bf Lemma 5.}
If $Y$ be a compact \Rs , then $L(Y)$ is discrete.
\end{lem}

\begin{rem}
 The statement is still true for surfaces of finite type but the proof is slightly more complicated and we restrict ourself to the compact case.
\end{rem}

\proof We realize $\pi _1 (Y)$ as a discrete group $\G$ acting in $\H ^2$
with the (relatively) compact fundamental domain $F$. Suppose that $Y$
contains infinitely many closed geodesics of the  length not greater than $C$.
 Then we can
lift them to segments  $[a_n , b_n ]$ in $\H ^2$ which intersect
the domain $F$. Then $\g _n (a_n ) = b_n$ for  some (different)
elements of $\G$. Therefore, $\g _n (F)$ intersect the $C$-\nbd \
of $F$. This contradicts to discreteness. \qed



\bigskip
\begin{thm}
%{\bf Theorem 8.}
There is a finite number $\g _j$ of elements of $\pi _1 (X)$
 such that any point $(Y, f) \in T(X)$ is determined by their length spectrum.
\end{thm}

\begin{rem}
 $Y$ isn't determined by $L(Y)$ as it was shown by M.F.Vigneras.
\end{rem}
\proof  We identify each $(Y, f)$ with the conjugacy class of
 admissible representation
$$
\rho : \G = \pi _1 (X) \to PSL(2, \R )$$
Then
$$
Tr ^2 (\rho (\g )) = 4\cosh ^2 \frac{\L (\g )}{2}$$
{\bf Algebraic proof.} We will use the fact that  $\G$ (and thus $\rho$) can be lifted in
 $SL(2, \R)$. Our proof follows \cite{Magnus}. Let $g_1 ,..., g_n$ be
any system of generators of. We can assume that $\rho (g_1 )$ is the diagonal matrix with given eigenvalues $\a , \a ^{-1}$.
Denote by   $t_{ij}$ the trace of $\rho (g_i g_j )$, etc.
After conjugation we can assume that $\rho (g_2 )$ is the matrix:
$$
\left(\begin{array}{cc}
 r & rs -1 \ne 0 \\
 1 & s \end{array}
 \right).
$$
Then $t_1 = \a + \a ^{-1} , t_2 = r + s , t_{12} = \a r + \a ^{-1} s$.
From this linear system we can find $\a , r, s$. Now, let
$$
\rho (g_3 ) = \left(\begin{array}{cc} \b & \g \\
 \delta & \e \end{array}
  \right)$$
when $\b \e - \g \delta = 1$. Then we know that
$$
\a \b + \a ^{-1} \e = t_{13}$$
$$
r \b + \delta (rs -1) + \g + s\e = t_{23}$$
And using $t_{123}$ we can find the forth equation on $\b , \g , \e , \delta$.
One can check that from these equations we can find the coefficients of the matrix $\rho (g_3 )$. Therefore, it's enough to take as the finite subset of
$\G$:
$$
g_1 ,..., g_n , g_1 g_k , g_2 g_j , g_1 g_2 g_s$$

{\bf Geometric proof.}

I recall that last quarter we proved that for every triple
$(2a, 2b, 2c) \in (\R _+ )^3$ there exists a hyperbolic pair of pants $P$ with
the lengths of boundary loops given by this triple. Now we want to prove
that $P$ is unique.

We split $P$ into union of 2 ``all right'' hexagons $X_1 , X_2$. Denote
by
$$
a_1 , \a , b_1 , \b , c_1 , \g$$
 the lengths of edges of $X_1$, then there is the hyperbolic cosine
formula relating  these numbers:
$$
\cosh c_1 \sinh a_1 \sinh b_1 = \cosh a_1 \cosh b_1 + \cosh \g$$
For proof see \cite{Beardon}.

This means that $\a , \b , \g$ determine $a_1 , b_1 , c_1$, thus
$a= a_1 , b= b_1 , c= c_1$. On another hand, $X_j$ is uniquely
determined by $a , b , c$. This proves that $P$ is uniquely
determined by $a , b , c$. \qed

Let $a_j , b_j$ , $j = 1,..., g$ be the canonical basis
of $\G$, $  d_i$ be as on Figure 8.


 \begin{figure}[tbh]
%\leavevmode
\centerline{\epsfxsize=3in \epsfbox{fig8.eps}}
\caption{}
%\label{fig:cyclic m-fold covering}
\end{figure}


We shall assume that the traces of the elements above and their double and triple products   are preserved by the representation $\r$.

We can assume that $\L (\rho )$ is the same as $\L (id )$ and
$$
\r (a _1 ) = a_1 , \r (d_1 ) = d_1 , \rho (a_1 d_1 ) = a_1 d_1$$
 since  the surface $X$ contains a pair of pants corresponding to
 $a_1 , d_1 , a_1 d_1$.
Now, the image $\r (a_2 )$ is obtained by conjugating $a_2$ via some isometry
$g$ which commutes with $d_1$ (since we can consider now the pair of pants corresponding to $a_2 , d_1$). Denote by $\a , \b , \g , \delta$ the axes of
the elements $a_1 , a_1 d_1 , a_2 , d_1$ in $\H ^2$. Then $g$ is a translation along $\delta$. We have to have: $dist (\a , \g ) = dist (\a , g\g )$ and
$dist (\b , \g ) = dist (\b , g\g )$ since the restrictions of $\r$
on $<a_1 , a_2 >, <a_1 d_1 , a_2 >$ are conjugations. But this means that
$g = 1$, since we  have to have: if $g$ isn't trivial then
it's a symmetry in the geodesic which is orthogonal to $\delta , \a , \b$.


So, we conclude that $a_2 = \r a_2$. The same argument can be applied to $b_2$.
Now, we can use the fact that $a_2 , d_1$ are fixed by $\r$ to prove that the element $b_1$ is fixed by $\r$ (applying the same arguments as above).
We can continue this process to prove that all $a_j , b_j$ are fixed by $\r$.

\qed

\begin{rem}
 It is known that for closed manifolds of nonpositive
curvature the
equality of marked length spectrums is equivalent to the existence of time preserving conjugation of the geodesic flows. In the dimension 2 it was recently proven by J.-P. Otal and C.Croke that surfaces of nonpositive curvature are uniquely determined by their length spectrums. In higher dimensions this is an outstanding research problem. It was announced by U.Hamenstadt that, if $M$, $N$ are closed manifolds of nonpositive curvature with conjugate geodesic flows so that $N$ has constant sectional curvature,  then $M, N$ are isometric.
\end{rem}

We shall need the following fact:

\begin{thm}
%Theorem 9}
\label{FK}
(See for instance \cite{FK}). If $G \subset PSL(2, \R )$ is a discrete group then the area
of fundamental domain of $\H ^2 /G$ is bounded from below by $\pi /21$.
\end{thm}

Actually, for us it will be enough to now the existence of some nonzero lower bound which we shall prove later.


\begin{cor}
 The order of the group of conformal automorphisms of any Riemann surface $Y$ of genus $g$ is note greater than $42(2g -2) = 84(g-1)$.
\end{cor}

\proof The area of $Y$ is $2\pi (2g- 2)$. This and Theorem
\ref{FK} imply Corollary. \qed

\begin{cor}
\label{finite}
 The kernel of the action of $Mod_X$ on $T(X)$ is finite.
\end{cor}

\begin{rem}
 Actually, the kernel is nontrivial only if $g = 2$ in which case the kernel is $\Z _2$. For any generic surface $Y$ of genus $> 2$ the group
$Aut(Y)$ is trivial.
\end{rem}

\begin{lem}
\label{stab}
 Let $[Y, f] \in T(X)$ have the stabilizer $H$ in $Mod_X$. Then
$H$ is isomorphic to the group of conformal automorphisms of $Y$.
\end{lem}
\proof Let $h \in H$, then  there is  a conformal automorphism
$c_h$ of $Y$ such that $f \circ h$ is homotopic to $c_h \circ f$.
Thus, we have a homomorphism $c: H \to Aut (Y)$. This homomorphism
is injective since the only element of $Aut(Y)$ homotopic to $id$
is $id$; and it is onto since for any automorphism $a \in Aut (Y)$
defines a homeomorphism $h$ of $X$ by the formula $h = f^{-1} a
f$. \qed

Now we can start the proof of Theorem \ref{Theorem 7}. Suppose that there exists a sequence
$g_n \in Mod_X$ such that
$$
\lim_{n \to \infty} g_n [X, id] = [Y, f]$$
Then, the corresponding monodromy representations in
$$
Hom (\pi _1 (X), SL(2, \R ))/SL(2, \R )$$ are convergent.
Therefore, the length spectrum of $g_n [X, id] = [X, (g_n )^{-1}]
= p_n$ is convergent to the length spectrum of $[Y, f]$. But the
unmarked length spectrum of $g_n [X, id]$ is the same as $\L (X)$.
Therefore, the discreteness of $\L (X)$ implies that for each $\g
_j$ in Theorem 8 there exists a number $n_j$ such that for all $n
> n_j$ we have: $\L _{\g_j} (p_n ) = \L _{\g_j} ([Y, f])$.
Therefore, for all large $n$ we have: $\L (p_n ) = \L [Y, f]$,
therefore, $[X, id]$ is a fixed point of the sequence $g_n \in
Mod_X$. However, we know that the stabilizer of any point in
$T(x)$ is finite (Lemma \ref{stab}, Corollary \ref{finite}).
Therefore the sequence $g_n$ is finite. This contradiction proves
the Theorem. \qed

\begin{cor}
 The moduli space $M(X) = T(X)/Mod_X$ is Hausdorff.
\end{cor}



\section{Compactification of the moduli space}

Our first aim is to prove the Mumford's compactness theorem for the moduli space (which is the reminiscence of the Mahler's compactness criterion):

\begin{thm}
%{\bf Theorem 12.}
For any $\e > 0$ the subset of the moduli space $M(X)$  consisting of surfaces with the injectivity radius $\ge \e$ is compact.
\end{thm}

\begin{lem}
%{\bf Lemma 7.}
 Let $X$ be a  \Rs \ of finite type and $\a _2$ be a simple closed geodesic on $X$. Then for any geodesic loop $\a _1$ intersecting $\a _2$
we have:
$$
\exp (l(\a _2 )) \ge (\exp (2l(\a _1 ) + 1)/(\exp (2l(\a _1 ) - 1)^2$$
\end{lem}
\proof Let $X = \H ^2 /\G$. Then we can assume that $\a _2$ corresponds to the transformation
$$
\g _2 : z \mapsto \l ^2 z$$
where $l(\a _2 ) = 2 \log (\l ) > 0$ and the axis is $\ell = <0, \infty >$;
and $\a_1$ corresponds to
$$
\g _1 : z \mapsto \frac{(B-k)z + B(k-1)}{(1-k)z + (kB -1)}
$$
where $\g _1 $ has the fixed points $1, B$ and $l(\a _1 ) = \log (k) >0$. Since
$\a _1 $ intersects $\a _2$ we conclude that $B < 0$. We have:
$$
\g _1 (\infty ) = (B- k)/(1-k ) >0$$
and
$$
1 > \g _1 (0) > 0$$
Similarly, since  $g_1 (\ell ) \cap g_2 g_1 (\ell ) = \emptyset$,
$$
\g _2 \g _1 (0) > \g _1 (\infty ) > 0$$
which means:
$$
\l ^2 B(k-1)/(kB -1) > (B-k)/(1-k)$$
Therefore,
$$
-\l ^2 B(k-1)^2 > (B-k)(kB -1) > - k^2 B -B$$
and
$$
\l ^2 > (k^2 + 1)/(k-1)^2$$ \qed
\bigskip
\begin{rem}
 If $l(\a _2 )= l$ and $\a _1$ intersects $\a _2$ then $l(\a _1 ) \ge f(l)$, where
$$
\lim_{l \to 0} f(l) = \infty$$
\end{rem}

\begin{cor}
 Suppose that $\a _1 , \a _2$ are simple loops on $X$ such
that:
$$
l(\a _1 ) \le 1 , l(\a _2 ) \le 1$$
Then $\a _1 , \a _2$ are disjoint.
\end{cor}
\proof  If $\a _1 \le 1$ then the left side of the inequality (*)
is $> (e^2 + 1)/2 \ge e$. \qed

\bigskip
\section{Zassenhaus discreteness theorem.}

Let $G$ be a Lie group, $[,] : G\times G \to G$. It's easy to see that the first derivative of this map at the point $(e, e)$ is zero. Therefore, there
exists a compact \nbd \ $U$ of $e$ in $G$ such that $[,]$ is a contracting map.

\begin{thm}
%{\bf Theorem 10}
(Zassenhaus). Suppose that $\G$ is a discrete subgroup of $G$ and $x, y \in \G \cap U$. Then the group generated by
$x, y$ is nilpotent.
\end{thm}
\proof  Form the sequence $x_0 = x , x_n = [x_{n-1}, y]$. Then $x_n \in \G \cap U$ for all $n$ and $\lim_{n\to \infty} x_n = e$. Therefore, the discreteness of $\G$ implies that for sufficiently large $n$ we have $x_n = e$.

{\bf Application.} Consider the case $G = Isom (\H ^n )$. Then any infinite nilpotent subgroup of $G$ is almost Abelian.

\bigskip

Suppose that $G = Isom_+ (\H ^n )$, $K$ is the maximal compact subgroup
 of $G$, $X = \H ^n = G/K$. We can assume that $G$ has left-invariant Riemannian metric which is also right-invariant under $K$. Then we shall identify
$X$ with a Borel subgroup $P$ of $G$. We can assume that $U$ in the Zassenhaus theorem  is $\e$-\nbd $U_{\e}(1)$ with respect to the metric on $G$, we shall denote the number $\e$ by $\e _Z$ (Zassenhaus constant).

\begin{lem}
%{\bf Claim 1.}
 There exist numbers $\m < \e _1 < \e$ and an integer $N$ such that: if $g_1 ,...,g_k$ generate a discrete group $\G$ so that for
$x = K \in G/K = X$ we have
$d(x, g_j x) \le \m$ then:

(1) $K$ has a $\e _1 /2$--net of $N$ elements;

(2) each word $w= w(g_1 ,..., g_k )$ of the length $\le N$ has the property:
$d(x, wx) \le \e _1$;

(3) for each $w$ as above we have:
$$
w(U_{3\e _1}(1))w^{-1} \subset U_\e (1)$$
\end{lem}
\proof We start with $\e _1 = \e /5$. Then we can find $N$ such
that (1) is satisfied, then we can choose $\mu$ such that (2) is
correct. Now, $w$ above is the product $pk$ where $d(p, 1) \le \e
_1$ (according to (2)), therefore, if $\delta \in  U_{3\e _1}$
then we put $\tau = k\delta k^{-1}$ then $\tau \in U_{3\e _1}$
since the metric on $G$ is biinvariant under $G$, thus
$$
d(1, pk\delta k^{-1}p^{-1}) = d(1, p \tau p^{-1}) \le 2d(1, p) + d(1, \tau )
\le 5\e _1
$$

Denote by $\G '$ the subgroup of $\G$ generated by all elements
$\g \in \G \cap U_{\e}(1)$ (a priori this subgroup can be
trivial). Denote by
$$
\G = \bigcup_{j=1}^{\nu} \g _j \G '
$$
the coset decomposition of $\G$.

\begin{lem}
%{\bf Claim 2.}
 In the decomposition above $\nu < \infty$.
\end{lem}

\proof Let $\g _j = w = g_{i_1} .... g_{i_M}$ be a word of the length
$M > N$. Then $w = w_1 w_2 = w_3 w_4$ where $l(w_2 ) < l(w_4 ) \le N$,
$w_j = p_j k_j$ and $d(k_2 , k_4 ) \le \e _1$. Therefore,
$$
\delta = w_4 w_2 ^{-1} = p_4 k_4 k_2^{-1}p_2 ^{-1}$$
However, $d(1, k_4 k_2^{-1}) \le \e _1$. On another hand,
$$
\e _1 \ge d(w_j x , x) = d(p_j x , x)$$
Therefore, $d(1, \delta ) \le 3 \e _1$. This implies that
$$
w = w_3 \delta w_2 \ \ \ , l(w_3 w_2 ) < M$$
Consider the element
$$
(w_3 w_2 )^{-1} w = w_2 ^{-1} w_3 ^{-1} w_3 \delta w_2 = w_2
^{-1}\delta w_2$$ Then the property (3) implies that this element
belongs to $U_{\e}(1) \cap \G \subset \G '$. Thus, $w_3 w_2$ and
$w$ belong to the same coset (mod $\G '$), but the length of $w_3
w_2$ is strictly smaller than $M$. The induction argument thus
imply that for all cosets (mod $\G '$) we can find representatives
such $\g _j$ that $l(\g _j ) \le N$. This implies  that $\nu$ is
finite. \qed


However, according to Zassenhaus theorem, the group $\G '$ is almost Abelian,
therefore, $\G$ is a finite extension of an Abelian group.

Thus we proved

\begin{thm}
%{\bf Theorem 11.}
For each $\H ^n$ there exists a constant $\mu = \mu _n $ such that: for any $x \in \H ^n$ and any elements $g_1 ,..., g_k \in Isom (\H ^n )$ which generate a discrete group $\G$ and $d(x , g_j (x)) \le \m$  the group
$\G$ is almost Abelian and is a finite extension of a subgroup $\G ' \subset \G$
 which is generated by elements $\g \in U_{\e} (1)$-- Zassenhaus
$\e _Z$--\nbd of $e$ in  $SO(n, 1)$.
\end{thm}

\begin{rem}
 According to \cite{Martin} one can
take as $\mu$ the number:
$$
9^{-(2+ [n/2])}$$
\end{rem}

\begin{rem}
 Actually we proved that for any $ \e < \e _Z$ we can find
$ \mu (\e ) < \e$ (increasing function on $\e $)
such that in each discrete group $\G _{\m (\e )} (x)$ the almost Abelian subgroup $\G ' _{\e} (x)$ has finite index. Here $\G _{\m (\e )} (x)$ is any
discrete group generated by elements $g_j$ such that
$d(g_j (x), x) \le \m (\e )$ and $\G ' _{\e} (x)$ is the maximal subgroup in
$\G _{\m (\e )} (x)$ generated by elements in $U_{\e} (1)$. The function
$\mu (\e )$ is invertible and $\e = \e (\mu )$. Moreover,
$$
\lim _{\m \to 0} \e (\mu ) = 0$$
\end{rem}

\bigskip


\begin{cor}
 There is an increasing  function $f(\l )$ defined for all
$\l < \m$ such that:

(a) $\lim _{\l \to 0} f(\l ) = \infty$;

(b) for any  $h, g \in SO(n, 1)$ and $x \in \H ^n$ such that
$d(x, h_x ) \le \l < \mu $    such that $<g, h>$ is discrete and nonelementary, we have:
$$
d(x, gx ) > f(\l )$$
\end{cor}
\proof Let $\psi _R (\delta )$ be an increasing function such that:
$$
p U_{\delta} (1) p^{-1} \subset U_{\psi _R (\delta )}(1)
$$
for all $p \in P\cap U_R (1)$ and for fixed $\delta$
$$
\lim_{\e = \psi _R (\delta ) \to 0} R = \infty$$

 For given $R$ denote by $\delta _R$ the number
such that $\psi _R (\delta _R ) = \e _Z$-- Zassenhaus constant. Now, let $\l$ be $\mu ( \delta _R )$ where $\mu$ is the function from the {\bf Remark} above.

Then the  group $\G ' _{\e (\l )} (x)$ has finite index in
$\G = <h>$
and $g \G ' _{\e (\l )} (x) g^{-1} \subset U_{\e _Z } (1)$. Therefore, according to Zassenhaus Theorem, the group generated by $g$ and
 $\G ' _{\e (\l )} (x)$ is almost Abelian and elementary, and thus the group
$<g, h>$ is elementary.

The function $\l = \l (R)$ is increasing and we can find the inverse
$R= f(\l )$. This function has the property that if $d(x, gx) \le f(\l )$
then either $<g, h>$ isn't discrete or is elementary (property (b)).
We have to verify the property (a):
$$
\lim _{\l \to 0} f(\l ) = \infty$$ Really, $\mu (\l ) \to 0$ as
$\l \to 0$ and $R \to \infty$ as $\delta _R \to 0$. \qed



\bigskip



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Suppose now that $X$ is a hyperbolic of finite type with totally geodesic boundary and $\a \subset \D X$, $l(\a ) = l$. Then there are three functions
$A(l), W(l), L(l)$ such that:

\begin{lem}
%{\bf Lemma 8.}

(1) $\a$ has a regular \nbd ~ $U$ of the width $W(l)$;

(2) the length of the second (different from $\a$) component $\a ^*$ of $\D U$ is
$L(l)$ and the area of $U$ is $A(l)$;

(3) $\lim_{l \to 0} W(l) = \infty$, $L(l)^2 \le Area (X)^2 + l^2$,
 $\lim_{l \to 0} L(l) \le  Area (X)$.
\end{lem}
\proof  Let $Y$ be the double of $X$.  Then, according to Lemma 8
the geodesic $\a$ on $Y$ has the normal injectivity radius at
least $W(l)= f(l)/4$. If we lift $U$ in the hyperbolic plane, then
the preimage of $\b$ is a hypercycle $\tilde{\b}$ which makes the
angle $\psi$ with $\tilde{\a}$. Now the hyperbolic trigonometry
and integration in polar coordinates imply that:
$$
\cosh (W) = 1/\cos(\psi )$$
$$
A(l) = Area (U) = l\cdot \tan (\psi )$$
$$
length (\a ^* ) = l /\cos (\psi )$$
However, $Area (U) < Area (X)$. Thus,
$$
Area(U) /l = \sqrt{L^2 /l^2 -1}$$
and
$$
L^2 = Area (U)^2 + l^2 \le Area(X)^2 + l^2$$ \qed
\bigskip

\begin{thm}
\label{Theorem 12}
 Let $X$ be a surface of finite type with totally geodesic boundary. Then there exists
a geodesic decomposition of $X$ on pants such that the lengths of the decomposing loops are bounded from above by some constant $C$ depending only on the topology of $X$ and the lengths of boundary curves.
\end{thm}
\proof  If $X$ is a pair of pants then we are done. Let
$\a _1 ,..., \a _m$ be the boundary curves of $X$. Denote by $W_i$ the width of the regular collar around $\a _i$ given by the Lemma 8. Denote by $\e$ the minimum of all
$W_i /2$. denote by $C(X)$ the union of $\e$-collars of $\a _j$ and put
$X^* = X - C(X)$.
If we can find on $X^*$ a homotopically nontrivial simple loop $\a$ which isn't homotopic to boundary which has length less than $\e$, then we are done. Suppose that such loops do not exist. If two collars intersect then the distance between two loops $\a _j ^*$ and $\a _i ^*$ is zero and the length of these loops is bounded by $Area (X)+ length(\D X )$. Thus, we can find a nontrivial curve $\b$ on $X$ as on the Figure  below. So, we assume that the collar $C(X)$ of $\D X$ is embedded.  Cover $X^* = X - C(X)$ by a maximal set of disjoint discs $D(P_j , \e ) \subset X$. If the number of these discs is $n$ then their
joint area is at least $n\e ^2$ and every point $z \in X$ has the property:
$$
d(z, \D (X^*  - \bigcup _{i=1}^n D_i (z, \e )) \ < \e
$$
Thus $n < Area (X) /\e$.  Draw a graph ${\cal G}$ on $X^*$ by connecting any two points $P_i , P_j$ such that
$d(P_i , P_j ) \le 4 \e$.

\begin{lem}
%{\bf Claim.}
The graph ${\cal G}$ is connected.
\end{lem}

\proof  Suppose that ${\cal G}$ consists of two  components $Z_1 ,
Z_2$. Consider the $\e$- \nbd s $U_1 , U_2$ of the unions of discs
with centers at $Z_1 , Z_2$ respectively. Then $U_1 \cup U_2
\supset X^*$ which means that their intersection isn't empty. This
implies that there are vertices in $Z_1 , Z_2$ such that the
distance between them is $\le 4\e$. This contradiction shows that
${\cal G}$ is connected. \qed

Thus, for every $z, w \in X^* $ we have:
$$
d (z, w) \le 4n\e + l(\partial X^* ) \le Area(X)/\e +
l(\partial X^* ) \le$$
$$
 Area (X) + Area(X)/\e + length(\D X) = c$$
and  the diameter of $X^*$ is bounded by a constant $c$ which depends only on topology of $X$ and the length of $\D X$. If we have at least two different boundary components $\a ^* _j , \a ^* _i$ of $X^*$ then we connect them by a shortest arc $\g$ and thus find a loop
$$
\b = \a ^* _j \cdot \g  \cdot \a ^* _i \cdot g ^{-1}$$
which has bounded by $2c Area(X)$ length and not boundary-homotopic. If we have only one boundary component $\a ^* _j$ then  choose a shortest  arc  $\g$
with endpoints on $\a ^* _j$ such that $\g \in \pi (X^* , \D X) \ne 0$. The length of this arc is bounded by $2 diam (X)$ (use 2-sheeted covering over $X$
which has 2 boundary components). Let $\a ^* _1$ be one of components of
$\a - \g$. Then take the loop
$$
\b = \a ^* _1 \cdot \g  $$
 The length of $\b$ is again bounded and this loop is homologically nontrivial,
and therefore- nonparallel to the boundary.

\qed

\begin{cor}
 Suppose that the injectivity radius of a closed surface $X$ is bounded from below by $\e$. Then the diameter of $X$ is bounded from above by
$$
c(\e ) = Area (X)/\e $$
On another hand, the area of $X$ in general is growing exponentially with the growth of $diam (X)$.
\end{cor}

\bigskip
Now we can prove Theorem \ref{Theorem 12}. There are only finitely
many nonequivalent pants decomposition of $X$ (the number $\nu
(X)$ of trivalent graphs with $2g-2$ vertices.) We fix $\nu (X)$
decompositions of $X$ with hyperbolic metrics: $X_1 ,..., X_{\nu}
\in T(X)$. Let $Y$ be any point of the subset $M(X)_{\e}$ of
$M(X)$ where $RadInj(Y) \ge \e$. Then there is a marking $(Y, f)$
on $Y$ such that $d_T ([Y,f], X_i ) \le b(\e , g)$ where $g$ is
the genus of $Y$. Therefore, the set $M(X)_{\e}$ is bounded and
therefore- compact.

Another remark. Suppose that $S \subset M(X)$ doesn't belong to
any $M(X)_{\e}$ for any $\e > 0$. Then $S$ is unbounded. Really,
for any choice of canonical generators of $\pi _1 (Y)$ we will
have: $a_j \cap \g \ne \emptyset$ where $length (\g ) < \e$.
Therefore, $length (a_j ) \to \infty$ as $\e \to 0$. This means
that the sequence of surfaces is not relatively compact in $M(X)$.
\qed

\bigskip
\begin{cor}
 There exists a number $q > 0$ so that for each (in particular noncompact) hyperbolic surface $X$ there is a point $p$ such that $RadInj _p (X) \ge q$.
\end{cor}
\proof  Continuity method. Let $X_{\m}$ be the subset of all
points in $X$ where $2 RadInj \le \mu$-  Margulis constant. This
is a disjoint union of annuli which at worst can be tangent. In
the worst case they decompose $X$ to the union of pairs of pants.
Consider this worst case. Then in the universal covering we have a
union of hypercycles which are at worst tangent one to another.
Then we can find a disc $D$ in $\H ^2$ which doesn't intersect
interiors of hypercycles and tangent at least to three of them. I
claim that the diameter of this disc is bounded from below by some
universal constant. Suppose not, we can assume that the center of
this disc is the point $0$. Then our configuration has a limit
where hypercycles degenerate to some discs which intersect or
tangent the boundary of $\H ^2$. But these discs are at worst
tangent and do not contain $0$. \qed

\begin{thm}
\label{Theorem 13}
 Suppose that $X_n$ is a sequence  in $M(X)$ such that
$$
\lim_{n\to \infty} [RadInj (X_n ) = \e _n] \to 0$$
 Then the sequence $X_n$ is not relatively compact in the moduli space $M(X)$.
\end{thm}
\proof  Let  $\g _n \subset X_n$ be the sequence of geodesic loops
such that $l(\g _n ) = \e _n$. Then for any choice of canonical
basis of $\pi _1 (X_n )$ there is a loop $\a _n$ in this basis
which has nonzero intersection number with $\g _n$. Therefore $l
(\a _n ) \to \infty$ as $n \to \infty$. This means that the
sequence $X_n$ is divergent in $M(X)$ with respect to the
Teichm\"uller metric. \qed



\section{Fenchel--Nielsen coordinates on\\  Teichm\"uller space.}

Consider a closed hyperbolic \Rs \ $X$ and fix a pants decomposition $D$ of $X$. For each pair of pants $P_j$ in $D$ there are  3 boundary loops $C_{j1},
C_{j2}, C_{j3}$. We connect them by the disjoint oriented geodesic arcs $\a _{ji}$  orthogonal to $\D P_j$. Each arc has the end-point $\zeta_{ji}$. Suppose that $C$ is a common boundary loop for two pairs of pants $P_j$,
$P_{j'}$ and $\zeta _{ji}$, $\zeta _{j' i'} $ are end-points of the arcs
$\a , \a '$.  The loop $C$ is oriented, so we can define
$$
\theta _{ij} = 2\pi d(\zeta _{ji} , \zeta _{j'i'}) \l(C_{ji})$$
where we calculate the distance in the positive direction. The number $\theta _{ij}$ is defined (mod $2 \pi $) and is called the ``angle of gluing''. Therefore, we have a continuous function
$$
\hat{F} = (L, \Theta ) : T(X) \to \R  ^{3g-3} \times \S ^{3g-3}$$
which associates with a point $Y \in T(X)$ the logarithms of  geodesic lengths
of the loops $C$ and $\Theta$ consists of the coordinates $\theta _{ji}$.
This map is obviously continuous and onto.

\begin{lem}
 The map $\hat{F}$ is a covering.
\end{lem}
\proof  If $\hat{F} (p= [Y, h]) = \hat{F} (q= [Z, g])$ then the surfaces
$Y, Z$ are isometric as unmarked Riemann surfaces. Therefore, $q= f(p)$ where
$f \in Mod_X$. The element of the modular group $f$ must preserve $D$. Denote
the subgroup of $Mod_X$ which preserve $D$ by $Mod_X (D)$. Then, for each
$f \in  Mod_X (D)$ and each $p \in T(X)$ we have:
$$
\hat{F}(p) = \hat{F}f(p)$$
therefore, $\hat{F}$ is a covering with the covering group $Mod_X (D)$.
Let's describe this group. This group is isomorphic to $Z^{3g-3}$. Therefore,
the Dehn twists along the loops $C$ in the pants decomposition generate
$Mod_X (D)$.

Now, the lift of $\hat{F}$ to $\R ^{6g -6}$ is denoted by $F$ and
is a homeomorphism of $T(X)$ which is called the Fenchel- Nielsen
coordinates on the Teichm\"uller space. \qed
\bigskip

\section{Riemann surfaces with nodes.}


The Riemann surface with nodes is a complex space modelled on $\C$
and $\{ zw = 0\} \subset \C ^2$ subject to the following
topological restriction: each surface with nodes can be obtained
from a nonsingular \Rs $S^*$ by pinching to points some system of
simple disjoint nonparallel homotopically nontrivial loops. We can
consider any surface with nodes as a (disconnected in general )
\Rs of finite type with a finite number of punctures together with
the identification pattern of the punctures.  The space of \Rs
with nodes forms a compactification of the moduli space.

Consider now the space $\hat{M}(X)$ - the space of Riemann surfaces with nodes
obtained by pinching some loops on elements of $M(X)$. We already have the topology on  $M(X) \subset \hat{M}(X)$. The {\bf horocyclic \nbd } \ of a point
$S \in \hat{M}(X) - M(X)$ is defined as follows. If $A= \{\a _1 , \a _2 ,..., \a _q \}$ are loops on $X$ to be pinched on $S$ then enlarge $A$ to a pants decomposition $D$ of $X$. Convention: we shall think that the loops $\a _j$
have zero length on $S$. Then the base of topology at $S$ is given by
neighborhoods $U_{\e}$ which consists of \Rs with nodes $Y$ such that:

(1) for pinched loops $\a _j$ the angles of gluing are arbitrary;

(2) the differences between all other Fenchel-Nielsen coordinates of
$S, Y$ are less than $\e$.


\begin{thm}
%{\bf Theorem 14.}
The space $\hat{M}(X)$ with the topology defined above is
compact, Hausdorff and has countable base of topology.
\end{thm}
\proof  The proof of compactness just repeats the proof of Theorem
\ref{Theorem 12}. We'd like only to note that the Fenchel-Nielsen
coordinates ``extend'' to the compactification $\hat{M}(X)$, but
different stratums of $\hat{M}(X) - M(X)$ correspond to different
pant decompositions of $X$ (so we have several coordinate systems
which cover  $\hat{M}(X)$). Two other statements are obvious.
\qed

\bigskip
There are several ways to construct the complex structure on the space
$\hat{M}(X)$, one- using algebraic geometry \cite{Mumford},  another- using
Kleinian groups and automorphic functions \cite{Bers}.



\section{Boundary of the space of quasifuchsian groups}

I recall the definition of the Bers' embedding of the
Teichm\"uller space:

Let $X = \H ^2 /F$ is a compact surface of genus $g > 1$ , and given $[\mu ] \in T(X)$ we lift $\mu$ to the hyperbolic plane $\H ^2$ (the upper half plane) and extend by zero to the whole complex plane, denote by $\nu$ the result.
Take $f= f^{\nu} $ to be solution of the Beltrami equation with the complex
characteristic $\nu$ which fixes the points $0, 1, \infty$. The map $f$ is conformal in the lower half-plane $\H ^2 _*$ and we can take the \Sd
$S(f)$ of the restriction of $f$ to $\H ^2 _*$. Then
$S(f)  \in Q(X)$ which we identify with the space of $F$-invariant
holomorphic \qd on $\H ^2 _*$. The \q.c. map $f$ defines the representation
$\rho : F \to \sll$ so that $f(g(z)) = \rho (g) f(z)$ for all $g \in F$.
The correspondence $\Phi : [\m ] \mapsto S(f)$ is called the ``Bers' embedding'', its image is a bounded domain $D$ in $Q(X)$.

We can assume that $0, 1, \infty$ are fixed points of three elements
$g_0 , g_1 , g_{\infty}$ of the group $F$. Therefore the assumption
that $\rho (g _j ) $ has the fixed point $j$ ($= 0, 1 ,\infty$) gives us a
slice on a Zariski open subset of $Hom (F , \sll )$  to the projection
$\pi : Hom (F , \sll ) \to Hom (F , \sll )//\sll = R(F)$.

On another hand, for each $\f \in Q(X)$ we have the monodromy
homomorphism $\rho _{\f }$ of the Schwarzian equation $S(f) = \f$
on the lower half plane $\H ^2 _*$. The map $hol : Q(X) \to Hom (F
, \sll )$  defined by the formula $hol(\f ) = \rho _{\f }$ is a
holomorphic map of $Q(X)$ to $R(F)$. It's easy to show that the
restriction of this map to $D$ is an embedding. Really, suppose
that $\f , \psi \in Q(X)$ be such that $hol (\f ) = hol (\psi )$.
Then we have two holomorphic maps $f_1 , f_2 : \H ^2 _* \to \C$
which are extendable to the boundary of the half-plane  and the
restrictions of $f_1$, $f_2$ to $\bar{\R}$ coincide (because the
fixed points of $G= hol (\f )(F)$ are dense on $f_i (\bar{\R})=
\Lambda (G)$ -limit set of the group $G$. Therefore, $f_1 = f_2$
and $\f = \psi = S(f_i )$. Moreover, one can prove that the  map
$hol$   is injective immersion on the whole space $Q(X)$
(Poincar\'e's lemma). Moreover, the space $hol (Q(X))$ is an
complex-analytic subvariety  in $R(F)$ (i.e. is a solution of an
equation $H(z) = 0$ for a vector-function $H : R(F) \to \C
^{3g-3}$.

For each $\f \in D$ the image of the representation $hol(\f )$ is a
``quasifuchsian group'' $G$ , the limit set of this group is a topological circle in $\c$.
However, we will be interested in the image of the boundary of $D$ under
$hol$. This is a compact subset of $R(F)$ and the images of the representations
in $hol(\D D)$ are called $b$-groups (boundary groups). For each $\f \in \D D$ the
solution of the equation $S(f) = \f$ is injective holomorphic function in $\h^*$
(by the continuity reasons: uniform on compacts limit of injective functions
on $\h^*$ is again injective). Therefore, $f(\h^* ) = \Omega _0 (G)$ is invariant under $G$ which implies that $\Omega _0 (G)$ is a subset of the domain of discontinuity $\Omega (G)$. Thus, each $b$-group $G$ is Kleinian.
The  compactness of $X = \Omega _0 (G) /G$ implies that the boundary of
  $\Omega _0 (G)$ consists of limit points of $G$. On another hand,
$\Omega _0 (G)$ is $G$-invariant, thus the whole limit set of $G$ coincides with the boundary of $G$. Another remark is that $\Omega _0 (G)$ is simply-connected, and $\rho : F \to G$ must be an isomorphism
(since it is induced by the injective conformal conjugation $f$ on $\h^*$).

Now we have to understand the topology of other components of $\Omega (G)$
(if there are any !). I recall that the group $G$ is finitely generated, therefore  the Ahlfors' finiteness theorem can be applied to $G$ as follows:

\bigskip

\begin{thm}
%{\bf Theorem 15}.
(Ahlfors' finiteness Theorem.) The quotient $Y = \Omega (G) /G$ of any finitely generated Kleinian group $G \subset \sll$ consists of a finite union of Riemann surfaces  of finite conformal type. Each puncture $p$
on $Y$ corresponds to a parabolic element of $G$ (i.e.if you lift a loop
around $p$ to $\Omega (G)$ then it is stabilized by a cyclic parabolic
subgroup of  $G$.
\end{thm}

For proof see \cite{Ahlfors2}, \cite{Kra}.

\medskip
I don't have any time to discuss the proof of this central fact  of theory of Kleinian groups, hopefully we shall do it next year.

From now on we shall denote by $Y_0$ the quotient $\Omega _0 (G)/G$.

Suppose now that some component $O$ of $\Omega (G)$ is not simply connected.
Consider the projection $Z$ of $O$ to $Y$. Then $Z$ is a boundary surface of
the 3-manifold $M(G) = (\H ^3 \cup \Omega (G) )/G$ and the induced
homomorphism  $i : \pi _1 (Z) \to \pi _1 (M)$ isn't injective. Therefore,
according to Dehn's lemma, we can find a simple closed curve $\g$ on $Z$
which isn't trivial on $Z$ but bounds an embedded  disc  $B$ in $M(G)$. Therefore, the fundamental group of $M(G)$ (isomorphic to $G$) splits into
a nontrivial free product. But it contradicts to our assumption that
$F\cong G $
 is the fundamental group of the closed surface $X$. Therefore, all components of $\Omega (G)$ are simply-connected. Thus, if $Z$ is any component of
$Y$, then $\pi _1 (Z)$ is a finitely generated subgroup  of $\pi _1 (X)$.
Suppose that the fundamental group $H$ of some component $Z$ of $Y \ne Y_0$ has a finite index in $G$. Then $H$ has the same limit set as $G$ and has at least
two invariant components of the domain of discontinuity: $\Omega _0 , \Omega _1$. Thus, the manifold $(M(H) - Y) \cup (\Omega _0 /H ) \cup Z$ is compact and is properly
homotopy equivalent to $Z \times [0,1]$. Theorem of Stallings implies that
 $M(H)$ is homeomorphic to $Z \times [0,1]$ and there exists a quasiconformal
map $\psi : \H ^3 \to \H ^3$ conjugating $H$ to a Fuchsian group.
This homeomorphism extends to the boundary $\c$ of $\H ^3$.
 That contradicts to our assumption that $G$ is a boundary group.

On another hand,  we know that all subgroups of $\pi _1 (X)$ of infinite index are  free. Therefore, if $Z$ is any component of $Y - Y_0$, then $Z$ is noncompact and cusps of $Z$ correspond to parabolic elements of $G$.

{\bf Conclusion.} Suppose that $G$ has no parabolic elements. Then $\Omega (G)
= \Omega _0 (G)$, i.e. it's connected and simply connected.

Kleinian groups with such ``pathological property'' are called ``totally degenerate Kleinian groups''. Our next goal is to prove the existence of such monsters.

For each $\g \in F - \{1\}$ we consider the polynomial function $T_{\g} $ on the representation variety $R(F)$:
$$
T_{\g} ([\rho ]) = Trace ^2 (\rho (\g ))$$
Thus the subset $S_{\g}= T^{-1}_{\g} (4) \subset R(F)$ is a complex-analytic subvariety as well as  the preimage $hol^{-1} S_{\g}$. Now, consider
the set $E$ of all real rays $R$ with origin at $0$ in $Q(X)$  so that
$$
R \cap \cup_{\g \in F -\{1\}} hol^{-1} S_{\g} = \emptyset$$ Almost
every ray in $Q(X)$  belongs to $E$ and  for each $R \in E$ the
groups $G = hol (R \cap \D D )$ have no parabolic elements.
Therefore, ``almost every group'' $G$ on the boundary of
Teichm\"uller space $hol (D )$ is totally degenerate.


\section{Examples of boundary groups}

Let $\{ c_1 ,..., c_k \} = C \subset X$ be a union of simple closed disjoint  nonparallel geodesics  on $X$. Lift
$C$ to the universal cover $\Delta = \H ^2$ of $X$. Denote the preimage by
$L$. Now, consider the following equivalence relation on $\c$:

$x \sim y$ if and only if they belong to the closure of one and
the same geodesic in $L$.

It follows from theorem of C.Moore that the quotient $\c /\sim$ is
homeomorphic to the sphere $\S ^2$. The action of the group $F$ on
$\c$ projects to the action of a group of homeomorphisms $G$ on
$\S ^2$. It can be proven that this action is conformal in some
conformal structure on $\S ^2$, thus $G$ becomes a Kleinian group
and this is a $b$-group.

The limit set $\Lambda$ of the group $G$ is the projection of the
boundary  of $\H ^2$, obtained by `` pinching '' the geodesics in
$L$. Projections of these geodesics are fixed points of parabolic
elements of $G$. The group $G$ has simply-connected invariant
component $\Omega _0$ - projection of $\H ^2 _*$. There are also
some non-invariant  components of the domain of discontinuity;
$\Omega (G)/G$ is homeomorphic to $X \cup (X- C = X_1 \cup ...\cup
X_s )$, where the component $X= X_0$ is covered by $\Omega _0$.
This follows from the fact that $\S ^2 - L$ is equivariantly
homeomorphic to $\c - (\Lambda (F) \cup L)$. The curves $X_1 ,...,
X_s$ are obtained from $X$ by ``pinching'' along $C$. The limit
set of the group $G$ looks like an infinite union of ``bubbles'' :
boundary curves of the domains covering $X_1 ,..., X_s$, two
bubbles can be tangent at the fixed point of a parabolic element.

Now, let me try to give you  an idea how the action of a totally
degenerate group looks like. Let $\f$ be a quadratic differential
on $X$ so that horizontal trajectories of $\f$ are never periodic.
Lift the horizontal foliation of $\f$ to a ``foliation'' $\F$ on
$\Delta$. [Technically speaking, this is a foliation on a
complement to some discrete subset- preimage of zeros of $\f$].
Each leaf of $\F$ is a ``quasigeodesic''  or a ``quasigeodesic
ray'' in $\H ^2$ (it's located in a finite distance from a
geodesic)  so its closure on $\D \Delta$ consists of 2 or 1
points. Now, again consider the equivalence relation $\sim$:

two points $x, y$ on $\Delta \cup \D \Delta$ are equivalent  if they belong to two
leaves $L_1 , L_2$ of $\F$ so that $Cl(L_1 ) \cap Cl (L_2 ) \ne \emptyset$.

It turns out that the quotient of $\c$ by $\sim$ is again a
topological sphere and the action of $F$ projects to a topological
action $G$ on $\S ^2$. Under some choice of $\f$ this action is
conformal in a conformal structure on $\S ^2$. The discontinuity
domain of $G$ consists of a single simply connected component
$\Omega _0$ - projection  of $\H ^2 _*$. The limit set $\Lambda$
of $G$ looks like an infinite tree which isn't locally finite.
This tree is ``dual '' to the foliation $\F$. The points of
branching of this tree are projections of the singular points of
the foliation and they a dense on $\Lambda$.

An example of $\f$ can be given as follows. Let $h : X \to X$ is a
homeomorphism with the property: for any $\g \in F$ and for any $n
\in \Z - \{0\}$ the elements $\g , h_* (\g )$ are not conjugate in
$F$. Such maps are called irreducible. Let $h_0$ is the extremal
(Teichm\"uller) quasiconformal map in the homotopy class of $h$.
Then $\f$ the \qd ~ corresponding to $h_0$.




\begin{thebibliography}{BaBE}
\addcontentsline{toc}{section}{Bibliography}


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W.\ Abikoff, ``Real analytic theory of Teichmuller Spaces'',
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\bibitem[Ah1]{Ahlfors}
L.\ Ahlfors, ``Lectures on quasiconformal maps'', 1966.

\bibitem [Ah2]{Ahlfors2}
 L.\ Ahlfors, {\em Finitely generated Kleinian groups}, Amer. J.
Math. {\bf 86} (1964) 413-- 429 ; {\bf 87} (1965) 759.



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   A.\ F.\ Beardon, ``The geometry of discrete groups''. N.Y.-
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\bibitem[B]{Bers}
L.Bers, {\em On spaces of \Rs with nodes }, Bull. of AMS, v. 80 (1974),  N 6, p. 1219- 1222.



\bibitem[C]{Crash}
``A Crash Course in Kleinian Groups'',  Lecture Notes in Mathematics, Vol. 400, 1974.

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H.\ Farkas, I. \ Kra, ``Riemann surfaces'', Springer Verlag.

\bibitem[G]{Gardiner}
F.\ Gardiner, ``Quadratic differentials and Teichmuller theory'',
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\bibitem[Kra]{Kra}
I.\ Kra, ``Automorphic Forms and Kleinian groups'', Benjamin
Reading, Massachusetts (1972).

\bibitem[L]{Lehto}
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Springer- Verlag, 1987.



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W.\ Magnus, {\em Monodromy of Hill's equations,} In: Collected Works,...

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G.\ Martin, {\em Balls in hyperbolic manifolds},
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B.\ Maskit, ``Kleinian groups''. Springer, 1987.

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C.\ McMullen, Amenability, Poincare' series and quasiconformal
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\end{thebibliography}

\end{document}
\end



\begin{thebibliography}{BaBE}
\addcontentsline{toc}{section}{Bibliography}


\bibitem[A]{Abikoff}
W.\ Abikoff, ``Real analytic theory of Teichmuller Spaces'',
Lecture Notes in Mathematics, Vol. 820, 1980.

\bibitem[Ah]{Ahlfors}
L.\ Ahlfors, ``Lectures on quasiconformal maps'', 1966.


\bibitem[Be]{Beardon}
   A.\ F.\ Beardon, ``The geometry of discrete groups''. N.Y.-
Heidelberg- Berlin: Springer, 1983.

\bibitem[FK]{FK}
H.\ Farkas, I. \ Kra, ``Riemann surfaces'', Springer Verlag.

\bibitem[G]{Gardiner}
F.\ Gardiner, ``Quadratic differentials and Teichmuller theory'',
1987.

\bibitem[L]{Lehto}
 O.\ Lehto, `` Univalent functions and Teichmullewr Spaces'',
Springer- Verlag, 1987.

\bibitem[M]{Maskit}
B.\ Maskit, ``Kleinian groups''. Springer, 1987.

\bibitem[N]{Nag}
S.\ Nag, ``Complex Analytic Theory of Teichmuller spaces'', 1988



\end{thebibliography}




\end{document}
\end


\section{Proper discontinuity of the mapping class group}

\begin{lem}
Let $S$ be a Riemann surface of finite hyperbolic type. Then the group of
conformal automorphisms $Aut(S)$ of the surface $S$ is finite.
\end{lem}
\proof Suppose that $g_n\in Aut(S)$ is an infinite sequence convergent to a certain $g\in Aut(S)$. Then for large $n$ the elements $g_n$ are homotopic
to each other. Recall that if $g\in Aut(S)$ is homotopic to the identity
then $g=id$. Thus all but finitely many elements in the sequence $g_n$ which shows that $Aut(S)$ is discrete. If $S$ is compact this immediately implies finiteness of $Aut(S)$. So we consider the case when $S$ is noncompact.
Lift $Aut(S)$ into the hyperbolic plane $\H^2$. The lift is a group $N$ which equals the normalizer of $\Ga=\pi_1(S)$ in $PSL(2,\R)$. The group $N$ is discrete since $Aut(S)$ is. Notice that $N/\Ga \cong Aut(S)$, thus our goal
is to show that $|N: \Ga|< \infty$. Consider the coset decomposition of $N$:
$$
N= g_0\Ga \u g_1\Ga \u g_2 \Ga ....
$$
where $g_0=1$. As we know, each discrete subgroup of $PSL(2,\R)$ has a fundamental domain, let $D$ be a (closed) fundamental polygon for $N$. Then
$$
P:= D \cup g_1 D \cup g_2 D ...
$$
is a fundamental domain for $\Ga$.
$$
\infty > Area(S)=Area(P)= \sum_i Area(g_i D)
$$
Since $Area(g_i D)= Area(D)$ we conclude that the sum is finite,
which in turn implies that $|N:\Ga|<\infty$. \qed

\medskip

\begin{thm}
The action of $Mod_S$ on $T(S)$ is properly discontinuous.
\end{thm}
\proof The Teichm\"uller space $T(S)$ is a proper metric space (metric balls in $T(S)$ are compact). Thus proper discontinuity of $Mod_S$ is equivalent to discreteness of $Mod_S$ in $Isom(T(S))$. Suppose that $Mod_S$ is not discrete. Then there exists a sequence of {\bf distinct} elements $[f_n]\in Mod_S$ such that $\lim_n [f_n] =[id]$. In particular, if $[S]$ is the origin in $T(S)$ then
$\lim_n [f_n]([S]) = [S]$, i.e. $\lim_n d([S,id], [S, f_n])=0$. Hence we can choose quasiconformal representatives $f_n:S\to S$ in $[f_n]$ such that
$$
\lim_n K(f_n) =1
$$
Each $f_n$ extends quasiconformally to the conformal compactification $\bar S$ of the surface $S$:
$$
\hat{f}_n: \bar{S}\to \bar{S}$$
and $K(\hat{f}_n)= K(f_n)$.

\begin{claim}
The sequence $\hat{f}_n$ is subconvergent to a conformal self-map of $\bar S$.
\end{claim}
\proof. There are several cases depending to the type of $\bar{S}$, I will consider only the cases when $\bar S$ is rational and hyperbolic, the elliptic case is left to the reader.

(a) Suppose that $\bar S$ is rational. Then $\bar S$ is the sphere with
 $p\ge 3$ punctures. Therefore $\hat{f}_n: \c \to \c$ is subconvergent on the set of at least 3 points (corresponding to the punctures). We let $\hat{f}$
be the limit of a subsequence, $f:= \hat{f}|S$. Then $K(f)=1$ and hence $f$ is a conformal automorphism of $S$.

\medskip
(b) Suppose that $\bar S$ is hyperbolic. Let $\t{f}_n$ be lifts
of $\hat{f}_n$ to the universal cover $\t{S}\cong \H^2$ of $\bar S$;
$\bar S= \H^2/\Ga$. We  retain the notation $\t{f}_n$ for the extension of $\t{f}_n$ to the closed hyperbolic plane $\H^2 \cup \D \H^2$.
The map $\t{f}_n$ conjugates the group $\Ga$ into itself. Pick a triple
of distinct points $x_1, x_2, x_3\in \D \H^2$. Then there exists a sequence $\ga_n \in \Ga$ such that
$$
\lim_n \ga_n \t{f}_n (x_j)= y_j
$$
(up to a subsequence) and the points $y_j, j=1,2,3$ are mutually distinct.
Therefore the sequence of quasiconformal maps $\ga_n \t{f}_n$ is subconvergent which implies that the sequence $f_n$ is subconvergent as well. Let $f: S\to S$  be the limit of a subsequence. Similarly to the case (a) this limit is a conformal self-map of $S$. \qed

\medskip
Now we can finish the argument. We choose a convergent subsequence in
$\{f_n\}$ (and retain the notation $\{f_n\}$ for this subsequence).
The maps $f_n$ are homotopic to the conformal map $f=\lim_n f_n$ for sufficiently large $n$. This contradicts the assumption that all the members of the sequence $[f_n]\in Mod_S$ are distinct. \qed
