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\raisebox{0in}[.5in][.2in]{
\textbf{MA 314 Homework 17: Building spaces by trading places} 
}
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These problems are intended to help improve your intuition for gluing polygons in euclidean, spherical and hyperbolic space.

\section{Mortar}
\begin{enumerate} \spacing
\item Summarize the major steps needed to prove that any surface created by gluing edges of polygons in pairs by isometries is locally isometric to $\E^2$, $\H^2$, or $S^2$ if the angle sum around each vertex is $2\pi$.

\item Give an example of polygons in $\E^2$ which glue up (by edge isometries) to be a sphere. What cone angles can you achieve?

\end{enumerate}

\section{Bricks}

\begin{enumerate} \spacing
\item  Suppose that $X \subset \E^2$ is a polygon without vertices (remember that the edges of $X$ have to be complete geodesics). Let $\ob{X}$ be obtained by gluing the edges of $X$ pairwise by isometries. What (topological) surfaces can you get? 

\item In the upper half-plane model of $\H^2 \subset \C$, let $X$ be the region bounded by the geodesics $E_1: x = -1 $ and $E_2: x = +1$. Let $\phi\co E_1 \to E_2$ be the map $\phi(z) = z + 2$. Note that $\phi$ is an isometry since it is a homothety. Let $\ob{X}$ be the glued up surface.
\begin{enumerate}\spacing
\item Explain why $\ob{X}$ is a cylinder (i.e. a bi-infinite annulus)
\item Let $P_1(y)$ and $P_2(y)$ be the points of $E_1$ and $E_2$ at height $y$. Observe that in $\ob{X}$, the points $\ob{P}_1(y)$ and $\ob{P}_2(y)$ are the same. 
\begin{enumerate}\spacing
\item What is the distance in $X$ between $P_1(y)$ and $P_2(y)$? (You may want to go back to chapter 2 of Bonahon to find an easy-to-use distance formula). Note that this distance is achieved by a hyperbolic geodesic $g(y)$ in $X$ which glues up in $\ob{X}$ to be a geodesic loop $\ob{g}(y)$ of the same length.
\item What happens to the length of the loop $\ob{g}(y)$ as $y$ approaches $0$? as $y$ approaches $\infty$?
\item For what value of $y$ is the length of the loop $\ob{g}(y)$ the smallest? the largest?
\item Use your answers to the previous questions to sketch a picture of the glued up cylinder $\ob{X}$ which captures something of its geometry.

\end{enumerate}
\end{enumerate}

\item In the upper half-plane model of $\H^2 \subset \C$, let $X$ be the region bounded by the geodesics $E_1: |z| = 1$ and $E_2: |z| = 2$. Let $\phi\co E_1 \to E_2$ be the map $\phi(z) = 2z$. Note that $\phi$ is an isometry since it is a homothety. Let $\ob{X}$ be the glued up surface.
\begin{enumerate}\spacing
\item Explain why $\ob{X}$ is a cylinder (i.e. a bi-infinite annulus)
\item Let $P_1(\theta)$ and $P_2(\theta)$ be the points of $E_1$ and $E_2$ an angle of $\theta$ from the positive $x$-axis. Observe that in $\ob{X}$, the points $\ob{P}_1(\theta)$ and $\ob{P}_2(\theta)$ are the same. 
\begin{enumerate}\spacing
\item What is the distance in $X$ between $P_1(\theta)$ and $P_2(\theta)$? (You may want to go back to chapter 2 of Bonahon to find an easy-to-use distance formula). Note that this distance is achieved by a hyperbolic geodesic $g(\theta)$ in $X$ which glues up in $\ob{X}$ to be a geodesic loop $\ob{g}(\theta)$ of the same length.
\item What happens to the length of the loop $\ob{g}(\theta)$ as $\theta$ approaches $0$? as $\theta$ approaches $\pi$?
\item For what value of $\theta$ is the length of the loop $\ob{g}(\theta)$ the smallest?
\item Use your answers to the previous questions to sketch a picture of the glued up cylinder $\ob{X}$ which captures something of its geometry.

\end{enumerate}
\end{enumerate}


\end{enumerate}






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