MAT 290-081: RFG "What is what in hyperbolic geometry"
Tentative time: Mondays,
15:10-16:00, MSB 1147.
Professor Michael Kapovich, MSB 2224.
Homepage: http://www.math.ucdavis.edu/~kapovich/RFG/rfg2010.html
First
meeting: Monday, January 10.
CRN for this class is: 30240
Credits: 1 credit for attending and 3 credits for giving a
talk.
The goal of this RFG
is to
be "broad and shallow", covering mostly definitions, concepts,
statements
and examples, rather than giving proofs. Almost all talks will be given
by students.
The topics will be:
Hyperbolic space: Upper half-space and unit ball models, angles,
volume form.
Isometry group of the hyperbolic space. Classification of isometries:
Hyperbolic, parabolic, elliptic.
Definition of hyperbolic manifolds. Hyperbolic structures on the pairs
of pants. Existence of hyperbolic structures on surfaces.
Riemann surfaces. Uniformization theorem.
Kleinian groups, quotient manifolds and surfaces.
Fundamental domains. Examples of Kleinian groups: Elementary, fuchsian,
quasifuchsian, Appolonian packings
Beltrami differentials and quasiconformal maps in 2d.
Extremal maps between Riemann surfaces.
Teichmuller spaces. Teichmuller space of the torus.
Mapping class group and discreteness of its action on the Teichmuller
space. Moduli space.
Fenchel-Nielsen coordinates.
Compactification of the moduli space.
Dehn twists. Classification of homeomorphisms of surfaces.
Geodesic laminations. Hausdorff topology on the space of geodesic
laminations.
Measured geodesic laminations and train tracks.
Length of a measured lamination and intersection numbers.
Topology of the space of measured laminations.
Measured foliations and their relation to measured laminations. Fat
train-tracks.
Harmonic maps from Riemann surfaces into trees.
Dual tree to a measured lamination. Skora's theorem: Converting trees
into laminations.
Thurston's compactification of the Teichmuller space.
Earthquakes and Nielsen realization problem/theorem.
Schwarzian differential equation and quadratic differentials.
Complex-projective structures and holonomy.
Relation of complex-projective structures to quadratic differentials
and measured laminations.
Ahlfors finiteness theorem.
Geometrically finite and infinite groups.
Pleated surfaces and how to construct them.
Ends of hyperbolic 3-manifolds, notion of tameness.
Tameness theorem. Ending laminations.
Model manifolds.
Ending Lamination Theorem.
Geometric and algebraic convergence.
Geometric structures on manifolds. Holonomy theorem.
3-dimensional geometries.
Geometrization Theorem.
McShane's Identity
Curve complex
Hyperbolicity of curve complex
References:
Wikipedea
M.Kapovich
"Hyperbolic
manifolds
and discrete groups"
C.McMullen
"Teichmuller
Theory"
J.Cannon
et
al
"Hyperbolic geometry"
M.Bestvina et al
"Convexity of hyperbolic length functions"
Y.Minsky
"Kleinian
groups"
W.Abikoff "Real-analytic theory of Teichmuller space"
Links to videos from MSRI workshops on Kleinian groups and Teichmuller
spaces:
General links:
https://secure.msri.org/calendar/workshops/WorkshopInfo/427/show_workshop
https://secure.msri.org/calendar/workshops/WorkshopInfo/428/show_workshop
Kleinian groups and hyperbolic manifolds:
Yair
Minsky
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 1
Yair
Minsky
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 2
Yair
Minsky
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 3
Kenneth
Bromberg
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 1
Kenneth
Bromberg
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 2
Kenneth
Bromberg
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 3
Jeffrey
Brock
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 1
Jeffrey
Brock
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 2
Jeffrey
Brock
, Introductory topics in Kleinian groups and hyperbolic
3-manifolds- Part 3
Anna
Wienhard
, Mostow rigidity- Part 1
Anna
Wienhard
, Mostow rigidity- Part 2
Teichmuller spaces:
Anna
Lenzhen
, Teichmuller Theory and its Metrics
Moon
Duchin
, The Curve Complex and its Relatives
Genevieve
Walsh
, Thurston's classification of surface automorphisms - Part 1
Genevieve
Walsh
, Thurston's classification of surface automorphisms - Part 2