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\begin{document}
\begin{center}
    Math 390C (Geometry in Group Theory)
    \\
    Daniel Allcock
    \\
    Homework 3, due Friday Feb 28, 2020
\end{center}

Everyone should solve and turn in
problems marked ``everyone''.  Undergrads
should also solve and turn in problems marked
``undergrads''.

The remaining problem is optional, but I found
it incredibly enlightening, so I highly recommend it.

Suppose $G$ is a group described by generators
and relations 
$$\gend{x_1,\dots,x_m:1=W_1(x_1,\dots,x_m)=
\cdots=W_n(x_1,\dots,x_m)}$$
Recall that
its \emph{Cayley graph} means the graph with
vertex set $G$, and a directed edge labeled `$x_i$'
from each $g\in G$ to $gx_i$.  If $x_i$ has order~$2$
then usually one draws a single unoriented edge rather
than arrows going both directions.  Usually one
uses colors for the edge labels.

\begin{problem}
    \undergrad
Construct a continuous family of right-angled
hexagons in the hyperbolic plane.  What is the
dimension of the space of all such hexagons up to
    isometry?  (I don't need any foundational
    dimension-theory
    stuff.  Just figure out how many parameters
    naturally describe a right-angled hexagon.)
\end{problem}

\begin{problem}
    \everyone
  Draw Cayley graphs for 
    \begin{align*}
        G&{}=\gend{x,y,z: 1=x^3=y^3=z^3=xyz}
        \\
        H&{}=\gend{x,y: 1=x^3=y^4=(xy)^4}
        \\
        I&{}=\gend{x,y: x^2=y^3}
        \\
        J&{}=\gend{a,b:aba=bab}
    \end{align*}
    Hint 1: there is an element of $Z(I)$
    visible; quotient by it first, build the
    Cayley graph for the quotient, and then
    use that to figure out that of $I$ itself.
    
    Hint 2: Find an isomorphism $J\iso I$.
    This gives directions for changing your graph
    for $I$ to one for~$J$.
\end{problem}

\begin{problem}
    \everyone
    Draw pictures of (or physically build) 
    the following 3-dimensional
    hyperbolic Coxeter polytopes:
    \begin{gather*}
    \bullet-4-\bullet-4-\bullet---\bullet
    \\
    \bullet\cdots\cdots\bullet-4-\bullet-4-\bullet-6-\bullet
    \end{gather*}
    The dots $\cdots$ mean that those faces of
    the polytope should be ultraparallel.

    Hint: these are nontrivial!  It might
    help to figure out the infinite-volume
    polytopes defined by only a subset of 
    (the hyperplanes containing) the walls.
\end{problem}

\begin{problem}
Let $G$ be the group 
$$
\gend{x,y,z:x^y=\xbar,y^z=\ybar,z^x=\zbar}
$$
Prove that it has a finite index subgroup 
    isomorphic to~$\Z^3$, and acts freely 
    on~$\R^3$.  (This means no element of 
    $G$ fixes any point of~$\R^3$, except that
    the
    identity element fixes every point.)
    Describe a fundamental domain for~$G$.
\end{problem}

\begin{problem}
    $\SU(1,1)$ is conjugate to $\SL_2\R$ in $\SL_2\C$.
    ($\SU(p,q)$ is the subgroup of
    $\SL_{p+q}\C$ that preserves a Hermitian inner product
    of signature $(p,q)$.  The standard example of
    such an inner product is
    $$
    \ip{x}{y}=x_1\ybar_1+\cdots+x_p\ybar_p
    -x_{p+1}\ybar_{p+1}-\cdots-x_{p+q}\ybar_{p+q}
    $$
    (Hint: What locus in $\C P^1$ do the norm~$0$
    vectors describe?)
\end{problem}

\end{document}
