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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due March 15 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 481 $|$ Topics in Geometry and Topology
    $|$ Spring 2012 $|$ M.~Macauley}}


\begin{document}

$\;$

\emph{Read}: Stahl, Chapters 9.1, 9.2.

\begin{enumerate}

\item Prove that the rotations $R_{0,\alpha}$ and $R_{0,-\alpha}$ are
  conjugate in $\Isom(\E^2)$.


\item Prove that the glide reflections $\gamma_{AB}$ and $\gamma_{CD}$
  are conjugate in $\Isom(\E^2)$ if and only if $d(A,B)=d(C,D)$.


\item Let $f=\gamma_{AB}$, $g=\gamma_{AC}$, where $A$, $B$, and $C$
  are distinct points of $\E^2$. Describe the Euclidean isometry
  $fgf^{-1}$ in standard form (i.e., as a translation, rotation,
  reflection, or glide-relection).


  % Stahl Exercise 9.2.1 (ii)(iv), p. 122
\item Find the image of the points $i$, $1+i$, and $-3+4i$ when
  subjected to the following isometries, where $0=(0,0)$ and $A=(3,0)$.
  \[
  \mbox{(i)}\quad I_{A,2}\hspace{1.5in}\mbox{(ii)}\quad
  f(z)=\frac{2z-1}{z+2}\,.
  \]


  % Stahl Exercise 9.2.2 (i)(iii), (old iv) p. 122
\item Express the following compositions as M\"obius transformations,
  where $0=(0,0)$ and $A=(3,0)$.
  \[
  \mbox{(i)}\quad I_{A,2}\circ I_{0,3}\hspace{0.7in}
  \mbox{(ii)}\quad\frac{2(-\bar{z})-1}{(-\bar{z})+2}\circ I_{0,3}
  \hspace{0.7in} \mbox{(iii)}\quad
  \frac{2(-\bar{z})-1}{(-\bar{z})+2}\circ\frac{2z-1}{z+2}\,.
  \]


\item Consider the following subgroups of $\GL_2(\R)$, the degree-$2$
  \emph{general linear group}: $\GL_2^+(\R)$ consists the matrices
  with positive determinant, and $\SL_2(\R)$, called the \emph{special
    linear group}, consists of the matrices with determinant $1$. The
  \emph{projective} versions of these groups are obtained by taking
  the quotient with their respective centers (scalars of the identity
  matrix). That is,
  \[
  \PGL_2(\R)=\GL_2(\R)/\<cI\>\,,\qquad
  \PGL^+_2(\R)=\GL^+_2(\R)/\<cI\>\,,\qquad
  \PSL_2(\R)=\SL_2(\R)/\{\pm I\}\,.
  \]
  \begin{enumerate}
    \item Prove that the center of the general linear group,
      $Z(\GL_2(\R))=\<cI\mid c\in\R^\times\>$, is a subset (and hence
      a a subgroup) of $\GL_2^+(\R)$.
    \item Prove that $\PGL_2^+(\R)$ is isomorphic to
      $\PSL_2(\R)$. [\emph{Hint}: Use the first isomorphism theorem --
      if $\varphi\colon G\to H$ is a homomorphism, then
      $G/\ker\varphi\cong\im\varphi$.]
  \end{enumerate}

\end{enumerate}

\end{document}


