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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due April 5 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}

\item Let $f$ be an orientation-reversing element of $\Isom(\H^2)$
  such that $\tau(f)=0$.
\begin{enumerate}
  \item Prove that $f$ is conjugate to the map $z\mapsto -\bar{z}$.
  \item Prove that if $r$ is a hyperbolic reflection, then $\tau(r)=0$. 
    % [\emph{Hint}: Let $z=x+yi$ and note that hyperbolic reflections
    % fix points in $\H^2$.]
  \item Prove that all hyperbolic reflections are conjugate. 
\end{enumerate}


\item Let $f$ be an orientation-reversing element of $\Isom(\R^2)$
  such that $\tau(f)<0$.
\begin{enumerate}
\item Prove that $f$ is conjugate to the map $z\mapsto -a\bar{z}$ for
  some $a>1$.
\item Let $g$ be another orientation-reversing element of
  $\Isom(\H^2)$ such that $\tau(g)<0$. Prove that $f$ and $g$ are
  conjugate if and only if $\tau(f)=\tau(g)$.
\item Prove that $f$ is the product of a hyperbolic reflection $f$ and
  an isometry $t$ of hyperbolic type such that $tr=rt$. Analogous to
  the Euclidean case, isometries of this type are called
  \emph{hyperbolic glide-reflections}.
\end{enumerate}


\item Consider a hyperbolic isosceles right triangle with hypotenuse
  of length $y$ and other sides of length $x$, and consider $y$ as a
  function of $x$.
  \begin{enumerate}
    \item Find a formula relating $x$ and $y$, and using implicit
      differentiation, find a formula for $y'$ as a function of $x$
      and $y$.
    \item Prove that $\sqrt{2}<y'<2$ for $x>0$, that
      $\displaystyle\lim_{x\to 0}y'(x)=\sqrt{2}$, and
      $\displaystyle\lim_{x\to\infty}y'(x)=2$.
  \end{enumerate}


\item This problem proves our third ``Thin Triangles Theorem'' for
  hyperbolic triangles.
  \begin{enumerate}
    \item Consider a hyperbolic isoceles right triangle with two sides
      of length $x$, and the hypotenuse of length $y(x)$. Using the
      results of the previous problem and differential calculus, prove
      that $y''>0$ for $x>0$.
    \item Now, consider a fixed hyperbolic isoceles right triangle
      with two sides of length $b$ and hypotenuse of length $a$. For
      $x\leq b$, let $y(x)$ be as before. Find and prove a
      relationship between $a/b$ and $y/x$.
    \item How does your answer in Part~(b) compare to the Euclidean case?
  \end{enumerate}
  In the field of geometric group theory, this property of $\H^2$ is
  called CAT(0) (pronounced ``cat-zero''), and spaces with this
  property are said to be \emph{non-positively curved}. A similar but
  more general statements holds for arbitrary (non-isoceles)
  hyperbolic triangles.
  

%\item Consider the hyperbolic quadrilateral $ABCD$ with $h(C,D)=y$,
%  $h(A,B)=z$, $h(A,D)=h(B,C)=x$, and right angles at vertices $C$ and
%  $D$, as shown below.
%    \begin{center}
%      \includegraphics[height=1in]{4gon}
%    \end{center}
\item Consider the hyperbolic quadrilateral $ABCD$ with angles
  $\alpha$, $\beta$, $\gamma$, and $\delta$, and side-lengths
  $h(C,D)=y$, $h(A,B)=z$, $h(A,D)=h(B,C)=x$. Moreover, suppose that
  $\gamma=\delta=\pi/2$.
  \begin{enumerate}
    \item Prove that $\alpha=\beta<\pi/2$, and that $z>y$.
    \item Now, let $f\in\Isom(\H^2)$ be an isometry of hyperbolic
      type. Prove that there is a uniue geodesic $q$ such that
      $f(q)=q$, setwise, and prove that $q$ is the set of minimal
      motion of $f$.
  \end{enumerate}


\item Consider the vertical lines $L_1$ and $L_2$ in $\H^2$ that have
  equations $x=-1/2$ and $x=1/2$, respectively, and let
  $\omega=(-\frac{1}{2},\frac{\sqrt{3}}{2})\in\H^2$.
  \begin{enumerate}
  \item Find a parabolic $p\in\Isom(\H^2)$ such that $p(L_1)=L_2$. 
  \item Suppose $f\in\Isom(\H^2)$ is an orientation-preserving
    isometry such that the order of $f$ is $3$. Prove that
    $\tau(f)=1$.
  \item Find an orientation-preserving $f\in\Isom(\H^2)$ such that
    $f(\omega)=\omega$ and the order of $f$ is $3$. [\emph{Hint}:
    Normalize $f$ to make $\det f=1$, and justify why you can do
    this.]
  \end{enumerate}

\end{enumerate}

\end{document}
