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\title{Homework 3 --- Hyperbolic geometry continued}

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

\maketitle

\begin{prob}
Recall the definition of the Bernoulli numbers: they are given by the
Taylor series expansion
$$\frac z {e^z - 1} = \sum_{n=0}^\infty \frac {B_n} {n!} z^n$$
If $\Lambda(\theta)$ denotes the Lobachevsky function, prove the identity
$$\Lambda(\theta) = \theta\Biggl(1 - \log |2\theta| +
\sum_{n=1}^\infty (-1)^{n+1} \frac {B_{2n}} {2n}
\frac {(2\theta)^{2n}} {(2n+1)!} \Biggr)$$
Hint: integrate $\cot(z)$ twice.

Use this formula to find an approximation to the volume of a regular
ideal tetrahedron. How many terms do you need to get $4$ decimal places?
\end{prob}

\begin{rmk}
With notation as above, $B_n = 0$ for $n$ odd and $>2$, so some authors use
the notation $B_n$ to denote the $2n$--th Bernoulli numbers, with the notation
above.
\end{rmk}

\begin{prob}
Using the notation
$$D(z) = \text{Im}(\text{Li}_2(z)) + \log|z|\arg(1-z)$$
and its interpretation as the volume of an ideal hyperbolic simplex with
parameter $z$, prove the {\em five term identity}:
$$D(x) - D(y) + D\Bigl(\frac y x\Bigr) - D\Bigl(\frac {1-x^{-1}} {1-y^{-1}}\Bigr) +
D\Bigl(\frac {1-x} {1-y}\Bigr) = 0$$ 
Hint: what are the cross--ratios of the 4--tuples from the set $(0,1,x,y,\infty)$?
\end{prob}

\begin{prob}
Let $H$ denote the quaternions, numbers of the form
$$h = a + bi + cj + dk$$
with $a,b,c,d \in \R$. For $h$ a quaternion, denote
$$h' = a - bi - cj + dk, \overline{h} = a - bi - cj - dk$$
Define $SB(2,H)$ to be the group of $2\times 2$ matrices
$\left[\begin{matrix} a & b \\ c & d\\\end{matrix}\right]$
with entries in $H$
such that $d=a',b=c'$ and $a\overline{a}-c\overline{c}=1$.

Consider the unit ball
$$B = \lbrace u = u_0 + u_1 i + u_2 j \in H \; | \; u_0^2 + u_1^2 + u_2^2 < 1 \rbrace$$
\begin{itemize}
\item{Show that the natural ``fractional linear'' action of $SB(2,H)$ on $B$ has image
exactly the group of orientation--preserving
conformal isomorphisms of the $3$--ball.}
\item{Show that this action of $SB(2,H)$ has kernel exactly $\pm \text{Id}$ 
and therefore that there is an isomorphism $SB(2,H)/\pm \text{Id} 
\cong PSL(2,\C)$.}
\end{itemize}
\end{prob}

\vfill
\pagebreak

\begin{prob}
The group $SU(2) \subset SL(2,\C)$ is the group of $2\times 2$
matrices such that
$\overline{A}^t A = \text{Id}$ and $\det(A)=1$. Define
$PSU(2) = SU(2)/\pm \text{Id}$ which can be thought of as a subgroup
of $PSL(2,\C)$. Show $PSU(2)$ can be identified with
the stabilizer of a point in $\H^3$. Deduce that $PSU(2) \cong SO(3,\R)$.
Give an explicit isomorphism $PSU(2) \to SO(3,\R)$.

The group $SO(3,\C)$ is the group of $3\times 3$ matrices $A$ such
that $A^t A = \text{Id}$ and $\det(A)=1$. Show that $PSL(2,\C) \cong SO(3,\C)$,
and give an explicit isomorphism between these groups
in such a way that the restriction to $PSU(2)$ induces the isomorphism in the
first part of the problem.
\end{prob}


\begin{prob}[Thurston]
Descriptions of spaces and geometries are often implicit. Manifolds often
arise in practice by parameterizing other geometric objects. The following
pretty example is due to Bill Thurston.

Consider the space of (not necessarily simple) pentagons in the plane having
$108^\circ$ angles and sides parallel to the corresponding sides of a model
regular pentagon, and parameterize this space by the (signed) side lengths
$s_1,\dots,s_5$.
\begin{itemize}
\item{Show that the $s_i$ are subject to a linear relation that confines them to
a three dimensional linear subspace $V$ of $\R^5$.}
\item{Show that the area enclosed by a pentagon is a quadratic form on $V$ which
is isometric to $E^{2,1}$. How do you measure area for a pentagon whose boundary
is not topologically a circle?}
\item{Describe a model for the hyperbolic plane in terms of the subset of $V$
consisting of pentagons of unit area. Does your model have a single component?}
\item{Show that the space of simple pentagons of unit area is a right--angled pentagon
in the hyperbolic plane.}
\item{There is an area--preserving ``butterfly operation'' on pentagons that
replaces a side of length $s$ by a side of length $-s$, changing the lengths
of the two neighboring sides to make it fit. Interpret this operation in
hyperbolic geometry.}
\item{Given a non--simple pentagon in $V$, when is it possible to modify it by
a sequence of butterfly moves until it is simple?}
\end{itemize}
\end{prob}

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