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\begin{document}
\begin{abstract}
This is a list of problems on hyperbolic geometry, complementing the problems in Problem Sets 4 and 5 and the lectures and discussions on the topic.\\
\end{abstract}

\title{MAT 141: Problems on hyperbolic geometry Set 1}

\maketitle

\vspace{-1cm}

\section{Problems on hyperbolic distances}

{\bf Problem 1}. For each of the following pairs of points $P,Q\in\H^2$, compute the hyperbolic distance $d_{\H^2}(P,Q)$:

\begin{enumerate}
    \item $P=3i$ and $Q=6i$.
    \item $P=2i+7$ and $Q=5i+7$.
    \item $P=i$ and $Q=1+2i$.
    \item $P=i$ and $Q=\rho+i$ where $\rho\in\R$ is any given real number.\\
\end{enumerate}

{\bf Problem 2}. For each of the following paths $\gamma$ from $P=i$ to $Q=1+i$, draw the image of the path $\gamma$ in $\H^2$ and compute their hyperbolic lengths $\ell_{\H^2}(\gamma)$:

\begin{enumerate}
    \item $\gamma(t)=t+i$, $t\in[0,1]$.
    \item $\gamma(\theta)=\frac{1}{2} + \frac{\sqrt{5}}{2}(\cos\theta + i\sin\theta)$, where $\theta\in[\arccos\left(-\frac{1}{\sqrt{5}}\right),\arccos\left(\frac{1}{\sqrt{5}}\right)]$.
    \item $\gamma(t) = t + i \left( 4\left(t - \frac{1}{2}\right)^2 \right)$, $t \in [0, 1]$.
    \item Which of the above three paths has minimal hyperbolic length?\\
\end{enumerate}

{\bf Problem 3} Consider the point $i\in\H^2$.

\begin{enumerate}
    \item Show that the set of points in $\H^2$ at hyperbolic unit distance from $i$ is
$$S_i:=\{z\in\H^2: |z-i\cosh(1)|=\sinh(1)\}.$$

\item Draw the set $S_i$ qualitatively in $\H^2$, explain why it is a circle, find its center and its radius.
\end{enumerate}

\section{Problems on hyperbolic lines}

{\bf Problem 4}. Consider $1+i\in\H^2$ and $L:=\{z\in\H^2:\mbox{Re}(z)=1\}$ a hyperbolic line containing it. Given the point $i\in\H^2$ outside $L$, show that there are infinitely many hyperbolic lines through $i$ that are parallel to $L$.\\

{\bf Problem 5}. Consider two points $P=iy_1$ and $Q=iy_2$ in $\H^2$, $y_1,y_2\in\R_{>0}$ and $y_1<y_2$.

\begin{enumerate}
\item Show that the unique hyperbolic line $E(P,Q)\sse\H^2$ given as the set of equidistant points to $P$ and $Q$ is
$$E(P,Q)=\{z\in\H^2: |z|^2=y_1y_2\}.$$

\item Describe explicitly, using a formula, a hyperbolic isometry $f:\H^2\lr\H^2$ whose fixed point set coincides with $E(P,Q)$.\\
\end{enumerate}

{\bf Problem 6}. Consider the two hyperbolic lines $L_1:=\{z\in\H^2:|z-2|=2\}$ and $L_2:=\{z\in\H^2:|z+2|=2\}$. Let $\rho\in(0,1]$ be a given real number.

\begin{enumerate}
\item Find the coordinates of the point $z_1(\rho)\in L_1$, given as the unique point in $L_1$ with $x$-coordinate equal to $\rho$. Similarly, find the coordinates of the point $z_2(\rho)\in L_2$ given as the unique point in $L_2$ with $x$-coordinate $-\rho$.\\

\item Show that the hyperbolic distance between $z_1(\rho)$ and $z_2(\rho)$ is $\ln\left(\frac{2+\sqrt{\rho}}{2-\sqrt{\rho}}\right)$.\\

\item Conclude that two hyperbolic lines sharing a common vertex in $\R\sse \partial\overline{\H}^2$ get closer and closer to each other in Euclidean distance with a rate of $\sqrt{\rho}$ when $\rho\to0$.
\end{enumerate}

{\bf Problem 7}. Consider the three hyperbolic lines $L_1:=\{z\in\H^2:|z-1|=1\}$, $L_2:=\{z\in\H^2:|z+1|=1\}$ and $L_3:=\{z\in\H^2:|z|=2\}$. Show that the area of the hyperbolic triangle bounded by $L_1,L_2,L_3$ is $\pi$.

\section{Problems on hyperbolic isometries}

{\bf Problem 8}. Consider the two maps $f_1,f_2:\H^2\lr\H^2$ given by
$$f_1(z)=-\overline{z},\quad f_2(z)=\frac{1}{\overline{z}}$$

\begin{enumerate}
    \item Show that $f_1$ and $f_2$ are orientation-reversing hyperbolic isometries.\\

    \item Describe the fixed points of $f_1$ and the fixed points of $f_2$.\\

    \item Determine whether $f_1\circ f_2$ equals $f_2\circ f_1$ or not.\\

    \item Find all the fixed points of $f_1\circ f_2$.\\

    \item Show that $f_1\circ f_2$ is not the identity but $(f_1\circ f_2)^2$ is the identity.\\
\end{enumerate}

{\bf Problem 9}. Consider the map
$$f:\H^2\lr\H^2,\quad f(z)=\frac{3\overline{z}+5}{5\overline{z}+3}.$$

\begin{enumerate}
    \item Show that $f$ is an orientation-reversing hyperbolic isometry.\\

    \item Prove that $f$ has no fixed points.\\

    \item Find a hyperbolic line $L\sse\H^2$ invariant under $f$, i.e.~such that $f(L)=L$.\\

    \item Find an explicit formula for the unique hyperbolic isometry $g:\H^2\lr\H^2$ such that $f\circ g$ is the identity.\\

    \item Find the fixed points of $g$.
\end{enumerate}

{\bf Problem 10}. Consider the map
$$f:\H^2\lr\H^2,\quad f(z)=\frac{(\sqrt{3}-1)z + 2}{-z + (\sqrt{3}+1)}.$$

\begin{enumerate}
    \item Show that $1+i$ is the unique fixed point of $f$.\\

    \item Show that $f^6$ is the identity.\\

    \item Prove that $f$ is a hyperbolic rotation centered at $1+i$ and determine its angle.\\
\end{enumerate}

\end{document}