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\title{Homework 4 --- Hyperbolic geometry}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

This homework is due November 3rd in Kathy Paur's mailbox. There will be no
class that day, since I'll be talking at a conference at Columbia. Recall
that $D$ usually denotes the Poincar\'e disk model of hyperbolic space,
and that $\partial D$ denotes the {\em ideal boundary} --- otherwise
known as the {\em circle at infinity}.

\begin{prob}
Let $l_1,l_2$ be two intersecting lines, in the Klein disk model, and
let $S$ be the boundary circle. Let $t_1,t_2$ be the two (Euclidean)
tangents to
$S$ at the endpoints of $l_1$.
Show that $l_1,l_2$ are perpendicular if and only if the three
lines $l_2,t_1,t_2$ are either parallel, or intersect in a single point.
(Here we think of $l_2$ as the Eudlidean straight line extending the
hyperolic straight line also called $l_2$).

Deduce the theorem in Euclidean geometry that $t_1,t_2,l_2$ are parallel or
coincident if and only if $s_1,s_2,l_1$ are parallel or coincident, where
$s_1,s_2$ are the tangents to $S$ at the endpoints of $l_2$.

Hint: to find the hyperbolic angle between $l_1,l_2$, move over to
the Poincar\'e disk model, where apparent angles are equal to hyperbolic
angles.
\end{prob}

\begin{prob}
For four distinct 
points $v_1,v_2,v_3,v_4 \in \R\cup\infty$, define the {\em cross--ratio} to be
$$c(v_1,v_2,v_3,v_4) = \frac{(v_4-v_2)(v_3-v_1)} {(v_2-v_1)(v_4-v_3)}$$
If $\alpha \in PSL(2,\R)$ acting on $\R\cup\infty$ in the usual way, then show
$$c(v_1,v_2,v_3,v_3) = c(\alpha(v_1),\alpha(v_2),\alpha(v_3),\alpha(v_4))$$
\end{prob}

\begin{prob}
With notation as in the previous problem, if $v_1,v_2,v_4$ are in anticlockwise
circular order, show there is a {\em unique}
choice of $\alpha \in PSL(2,\R)$ with 
$$\alpha:\lbrace v_1,v_2,v_4 \rbrace \to \lbrace 0,1,\infty\rbrace$$
(the bijection induced by $\alpha$ should be the order--preserving one).
Show that $$c(v_1,v_2,v_3,v_4)=\alpha(v_3)$$
\end{prob}

\begin{prob}
In the upper half--space model, show that the set of points $P$ at constant
{\em hyperbolic} distance $t$ away from the hyperbolic straight line $x=0$ consists
of a pair of {\em Euclidean} straight lines. What is the relationship
between the distance $t$ and the angle between the lines in $P$ and the
line $x=0$?
\end{prob}

\begin{prob}
Let $K$ be the group of matrices of the form
$$\left[\begin{matrix}
\cos(\alpha) & \sin(\alpha) & 0 \\
-\sin(\alpha) & \cos(\alpha) & 0 \\
0 & 0 & 1 \\
\end{matrix}\right]$$
and $A$ the group of matrices of the form
$$\left[\begin{matrix}
\cosh(\gamma) & 0 & \sinh(\gamma) \\
0 & 1 & 0 \\
\sinh(\gamma) & 0 & \cosh(\gamma) \\
\end{matrix}\right]$$
Show that every element of $SO(2,1)$ can be expressed as
$k_1ak_2$ for some $k_1,k_2 \in K$ and $a \in A$; that is, we can write
$SO(2,1) = KAK$. How unique is such an expression?
\end{prob}

\begin{prob}
Let $K'$ be the subgroup of $PSL(2,\R)$ consisting of matrices of the form
$\left[\begin{matrix} \cos(\alpha) & \sin(\alpha) \\ -\sin(\alpha) & \cos(\alpha)
\\ \end{matrix}\right]$ and
$A'$ the subgroup of matrices of the form
$\left[\begin{matrix} s & 0 \\ 0 & s^{-1} \\ \end{matrix}\right]$. Find an
isomorphism from $SO(2,1)$ to $PSL(2,\R)$ taking $K$ to $K'$ and $A$ to $A'$.
(Careful! The isomorphism $K \to K'$ might not be the one you first think of . . .)
\end{prob}

\begin{prob}
Recall the subgroups $K'$ and $A'$ defined in the previous question. Let $N$ denote the
group of matrices of the form 
$\left[\begin{matrix}1 & t \\ 0 & 1 \\ \end{matrix}\right]$.
Show that every element of $PSL(2,\R)$ can be expressed as $kan$ for some $k \in K'$,
$a \in A'$ and $n \in N$. How unique is this expression? This is an example of what is
known as the {\em $KAN$} or {\em Iwasawa decomposition}.
\end{prob}

\begin{prob}
For a group $G$, the {\em commutator subgroup} $G_1$ is the group generated
by elements of the form $[g,h]$ for $g,h \in G$ (remember $[g,h] = ghg^{-1}h^{-1}$).
We write $G_1 = [G,G]$. Define $G_i$ inductively as the group generated by
elements of the form $[g,g_i]$ for $g \in G$ and $g_i \in G_{i-1}$. We write
$G_i = [G,G_{i-1}]$. A group $G$ is {\em nilpotent} (of order $n$) if
$G_n$ is trivial, for some $n$.

Show $N$ as in the previous problem is nilpotent (this motivates the standard
notation).
\end{prob}

\begin{prob}
We inherit notation from the previous problems.
In the Poincar\'e disk model, show that the orbits of points 
under the group $K$ are circles contained in the interior of $D$.
Show that the orbits of
points under the group $A$ are arcs of circles (or straight lines) which
intersect the boundary of $H$ transversely. Show that the orbits of points
under the group $N$ are circles which intersect $\partial D$ tangentially.
Use this to show that no element of one of these subgroups is a conjugate
of an element of another one of these subgroups unless they are the
identity elements.

These third kind of ``circles'' are called {\em horocircles} or sometimes
{\em horocycles}.
\end{prob}

\begin{prob}[Hard]
Let $\gamma$ be a smooth curve embedded in $\R^2$, bounding a region $E$.
Define a ``metric'' on $E$ as follows: Let $f$ be a smooth nowhere zero
function on $E$ which is equal to $\frac 1 {\text{dist}(p,\gamma)}$ for all
$p$ sufficiently close to $\gamma$. Let the metric on $E$ at a point $p$
be equal to the Euclidean metric at that point times $f(p)$.
That is, the ``length elements'' on $E$ are given by $(fdx,fdy)$, where
$(dx,dy)$ are the usual Euclidean length elements. Show that there is
a continuous $1$--$1$ map $\phi:E \to D$ which distorts the lengths of curves
by a bounded amount. Here $D$ is the Poincar\'e disk model of hyperbolic
space with the hyperbolic metric. That is, there is a constant $K>0$ such that
for any curve $\alpha$ in $E$,
$$\frac 1 K \text{length}_D(\phi(\alpha)) \le \text{length}_E(\alpha) \le
K \text{length}_D(\phi(\alpha))$$
\end{prob}

\begin{rmk}
One form of the Riemann mapping theorem says that there is a unique
conformal (i.e. angle preserving) map $\phi:D \to E$ which takes any three
distinct points in $\partial D$ to any three distinct points in $\partial E$.
If we define a metric in $E$ by declaring that the length of a segment
$\gamma$ in $E$ is equal to the hyperbolic length of $\phi^{-1}(\gamma)$,
then this metric is asymptotically equal to the metric defined in the
previous problem, for points close to the boundary.
\end{rmk}

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