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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due March 6 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl Chapters 6.1, 6.2, 7.1, 7.2, 8.1, 8.2, 8.3.

\begin{enumerate}

  % Stahl Exercise 6.6.8 (old version), p. 108
\item Prove that if $0<\alpha<2\pi$ then there is a hyperbolic
  quadrilateral whose angles sum to $\alpha$.


\item For $a,b>0$, let $R(a,b)$ be the Euclidean rectangle with
  corners $(0,2)$, $(a,2)$, $(0,2+b)$, and $(a,2+b)$.
  \begin{enumerate}
    \item Give a qualitative description of what $R$ looks like to a
      resident of the hyperbolic plane. Specifically, which of its
      sides are hyperbolically straight or hyperbolically curved. If
      curved, in which direction?
    \item For a given $a,b>0$, let $\ha(R(a,b))$ denote the hyperbolic
      area of $R(a,b)$, let $L(a,b)$ be the hyperbolic length of the
      left side of $R$, and let $B(a,b)$ be the hyperbolic length of
      the bottom side of $R$. Prove that
      \[
      \lim_{a,b\to 0^+}\frac{L(a,b)B(a,b)}{\ha(R(a,b))}=1\,.
      \]
  \end{enumerate}
  
\emph{For the next two problems, consider a hyperbolic triangle
  $ABC$ with right angle at $C$, let $\alpha$ and $\beta$ denote the
  angles at $A$ and $B$, respectively, and let $a$, $b$, and $c$ be
  the hyperbolic lengths of the sides opposite $A$, $B$, and $C$,
  respectively.}


\item Prove Parts (i)--(iii) of Proposition 8.2.2 in Stahl: If
  $\gamma$ is a right angle, then the following identities hold.
  \begin{center}
    \begin{tabular}{rlcl}
      i. & $\tanh a=\sinh b\tan\alpha$ & \quad and\qquad & $\tanh b=\sinh
      a\tan\beta$ \\ ii. & $\sinh a=\sinh c\sin\alpha$ & \quad
      and\qquad & $\sinh b =\sinh c\sin\beta$ \\ iii. & $\tanh b=\tanh
      c\cos\alpha$ & \quad and\qquad & $\tanh a =\tanh c\cos\beta$.
    \end{tabular}
  \end{center}
  Additionally, find the Euclidean analogues of these formulas, and
  prove that the hyperbolic versions approach the Euclidean versions
  in the limit as $a,b,c\to 0$.
  

% Stahl Exercise 8.3.17, p. 108
\item Theorem 8.3.2(i) in Stahl is the formula
  $\displaystyle\cos\alpha=\frac{\cosh b\cosh c-\cosh a}{\sinh b\sinh
  c}$. The proof of this began by considering $d$, the hyperbolic
  altitude from vertex $A$ to side $BC$. The case when $d$ falls
  inside the triangle was done in the book. Finish the proof of this
  statement when $d$ falls outside the triangle.


\item Recall that hyperbolic triangles are ``thin'' in the sense that
  the sum of their angles must be less than $\pi$. In this problem, we
  establish another ``thinness'' property of hyperbolic triangles: If
  $ABC$ is a hyperbolic triangle, then every point of $AC$ is at most
  distance $1$ from some point on $AB$ or some point on $BC$.
  \begin{enumerate}
  \item Prove the theorem in the case where either one of the angles
    at $A$ or $C$ is non-acute.
  \item Prove the theorem in the case where the angles at $A$ and $C$
    are both acute.
  \end{enumerate}

  
\item Let $G$ be a group. Prove that conjugacy is an equivalence relation. 


\item Let $X$ be a geometry and $G=\Isom(X)$. 
  \begin{enumerate}
  \item Let $x,y\in X$ and $f,g\in G$, and define $h=fgf^{-1}$. Prove
    that $g\colon x\mapsto y$ iff $h\colon f(x)\mapsto f(y)$. Draw a
    commutative diagram relating $f$, $g$, and $h$.
  \item Prove that if $x$ is a fixed point of $g$, then $f(x)$ is a
    fixed point of $h$.
  \end{enumerate}

  

\item Let $\tau_\v$ be the traslation by the vector $\v\in\E^2$.
  \begin{enumerate}
  \item Prove that the minimal motion of $\tau_\v$ is $|\v|$, and the
    set of minimal motion of $\tau_\v$ is all of $\E^2$.
  \item Prove the if $R$ is a rotation around the origin, then
    $R\tau_\v R^{-1}=\tau_{R(\v)}$. Draw a commutative diagram showing
    this relation.
  \end{enumerate}


\end{enumerate}

\end{document}
