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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due March 29 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl, Chapters 9.3, 9.4, 9.5.

\begin{enumerate}


\item Let $Z$ be the set of $2\times 1$ nonzero column vectors over
  $\C$, and let $\mathcal{Z}$ be the set of equivalence classes of $Z$
  generated by the equivalence relation that
  $\mathbf{w}\sim\mathbf{z}$ if $\mathbf{w}=\lambda\mathbf{z}$ for
  some $\lambda\in\C^\times$. Let
  $\Phi\colon\mathcal{Z}\mapsto\hat{\C}$ be defined by the rule
  \[
  \lambda\left(\!\!\begin{array}{c}z_1\\z_2\end{array}\!\!\right)
  \stackrel{\Phi}{\longmapsto}
  \left\{\!\begin{array}{ll}\frac{z_1}{z_2} & \mbox{if }z_2\neq 0
  \\ \infty & \mbox{otherwise.}\end{array}\right.
  \]
  \begin{enumerate}
    \item Show that $\Phi$ is bijective.
    \item The group $\GL_2(\R)$ acts on $\mathcal{Z}$ by the rule
      \[
      \A.\z=\left\{\begin{array}{ll}\A\z & \mbox{if $\det\A>0$}
      \\ \A\bar{\z} & \mbox{if $\det\A<0$,} \\ \end{array}\right.
      \]
      with the convention that
      $\displaystyle\frac{a\infty+b}{c\infty+d}=\frac{a}{c}$ and
      $\displaystyle\frac{a}{0}=\infty$. Show that
      \[
      \Phi(\A.\z)=\mu_A(\Phi(\z))\,,
      \]
      where $\mu$ is the M\"obius map. [\emph{Hint}: Consider
      $\z=\left(\!\!\begin{array}{c}z\\ 1\end{array}\!\!\right)$.]
  \end{enumerate}
  

  % Proof of Theorem 6.2.4
\item This problem completes the proof of the classification of
  hyperbolic isometries.
  \begin{enumerate}
    \item Show that $\ker\mu=Z(\GL_2(\R))$.
    \item Prove that $\Isom(\H^2)\cong\PGL_2(\R)$.
  \end{enumerate}



\item This problem proves that $\PGL_2(\R)$ acts $3$-transitively on
  ideal points, and is completely determined by where it sends $0$,
  $1$, and $\infty$.
  \begin{enumerate}
  \item Prove that there is an element of $\PGL_2(\R)$ which sends
    $(0,1,\infty)$ to any three distinct points $(a,b,c)$ of $\hat\R$,
    in that order.
  \item What can you say about an element of $\PGL_2(\R)$ that fixes
    $0$, $1$, and $\infty$?
  \item Prove that any elements of $\PGL_2(\R)$ is uniquely determined
    by where it sends $0$, $1$, and $\infty$.
  \end{enumerate}


\item Prove that every ideal triangle has area $\pi$. 


\item Let $f$ be an orientation-reversing element of $\Isom(\H^2)$
  such that $f(\infty)=\infty$.
\begin{enumerate}
  \item Prove that $f$ also fixes a point in $\R$.
  \item If we additionally know that $f(0)=0$, prove that
    $f(z)=a\bar{z}$ for some $a<0$.
\end{enumerate}


\item Let $f$ be an orientation-reversing element of $\Isom(\H^2)$. 
  \begin{enumerate}
  \item Prove that $f$ fixes two points in $\hat\R$.
  \item Prove that $f$ is conjugate to some orientation-reversing
    element of $\Isom(\H^2)$ that fixes both $0$ and $\infty$.
  \end{enumerate}
  
\end{enumerate}

\end{document}
