This is a LaTeX-file and in order to get the cross-references
right it has to be run twice.
BODY
\documentstyle[12pt,dina4p]{article}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\rto}{\rightarrow}
\newcommand{\lto}{\longrightarrow}
\newcommand{\mto}{\mapsto}
\newcommand{\vlto}{-\!\!\!-\!\!\!-\!\!\!\!\longrightarrow}
\newcommand{\kz}{I\!\!\!\!C}
\newcommand{\pz}{I\!\!P}
\renewcommand{\gz}{Z\!\!\!Z}
\renewcommand{\nz}{I\!\!N}
\renewcommand{\rz}{I\!\!R}
\newcommand{\qz}{I\!\!\!\!Q}
\renewcommand{\H}{I\!\!H}
\newcommand{\unmat}{1 \;\!\!\!\! 1}
\newcommand{\dcup}{\stackrel{.}{\cup}}
%
\def\3{\ss}
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cE{{\cal E}}
\def\cF{{\cal F}}
\def\cG{{\cal G}}
\def\cH{{\cal H}}
\def\cI{{\cal I}}
\def\cJ{{\cal J}}
\def\cL{{\cal L}}
\def\cM{{\cal M}}
\def\cN{{\cal N}}
\def\cO{{\cal O}}
\def\cQ{{\cal Q}}
\def\cR{{\cal R}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
\def\cV{{\cal V}}
\def\cW{{\cal W}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\la}{\lambda}
\newcommand{\om}{\omega}
\newcommand{\Om}{\Omega}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\ro}{\rho}
\newcommand{\ve}{\varepsilon}
\newcommand{\vp}{\varphi}
\newcommand{\vt}{\vartheta}
\newcommand{\lb}{\overline{\lambda}}
\newcommand{\ab}{{\al\be}}
\newcommand{\ag}{{\al\ga}}
\newcommand{\bg}{{\be\ga}}
\newcommand{\mn}{\mu\nu}
%
\def\R{\mbox{\makebox[.2em] [l] {I}R}}
\def\N{\mbox{\makebox[.2em] [l] {I}N}}
\def\Z{\mbox{\makebox[.22em] [l] {\sf Z}{\sf Z}}}
\def\C{\makebox[.6em]{\makebox [-.18em]{C}\rule{.1em}{1.5ex}}}
\def\Q{\makebox[.6em]{\makebox [-.25em]{Q}\rule{.08em}{1.5ex}}}
\begin{document}
\noindent DESY 92-104 \hfill ISSN 0418-9833\\
July 1992
\vspace*{3cm}
\begin{center}
{\LARGE\bf Periodic Orbits in Arithmetical Chaos} \\
\vspace*{4cm}
{\large Jens Bolte} \\
\vspace*{2cm}
II. Institut f\"ur Theoretische
Physik\\ Universit\"at Hamburg\\ Luruper Chaussee 149, 2000
Hamburg 50\\ Fed. Rep. Germany
\end{center}
\vfill
\begin{abstract}
Length spectra of periodic orbits are investigated for
some chaotic dynamical systems whose quantum energy spectra
show unexpected statistical properties and for which the
notion of arithmetical chaos has been introduced recently.
These systems are defined as the unconstrained motions
of particles on two dimensional surfaces of constant negative
curvature whose fundamental groups are given by number
theoretical statements (arithmetic Fuchsian groups).
It is shown that the mean multiplicity of lengths $l$ of
periodic orbits grows asymptotically like $c\cdot e^{l/2}/l$,
$l\rto \infty$. Moreover, the constant $c$ (depending on
the arithmetic group) is determined.
\end{abstract}
\newpage
%
\section{Introduction}
In the semiclassical analysis of quantum mechanical systems
one is interested in detecting traces of the properties of
the underlying classical systems. These may either be
integrable or totally chaotic, or their phase spaces
consist of regions that are in some intermediate stage. Many
investigators in the field for example have worked on the problem
how to distinguish statistical properties of
semiclassical energy spectra in the two extreme cases of
integrable and strongly chaotic classical systems. It has been
found \cite{Berry} that the existence of action--angle
variables for integrable systems leads to a Poissonian statistics
for the nearest--neighbour spacings distribution in the
(semiclassical) energy spectrum. (For rigorous results, see
\cite{Sinai}.) Concerning chaotic systems,
however, no such result exists, although it is generally believed
that the energy level statistics for classically chaotic
systems may be described by the distribution of eigenvalues
of large random hermitian matrices. This {\it random matrix
theory} (RMT) \cite{RMT} offers a phenomenological
description of energy spectra and has been confirmed
empirically in many examples \cite{GOE}.
Recently, a class of strongly chaotic systems has been found
with energy spectra that do not fit into the universal
scheme of RMT \cite{ArithmChaos}, but rather
appear to be more like ones of classically integrable systems.
These exceptional systems consist of single particles sliding
freely on special two dimensional hyperbolic surfaces,
i.e. surfaces endowed with a metric of constant negative
curvature. In more sophisticated terms, the classical
dynamics are defined by the geodesic flows on those surfaces.
One realizes hyperbolic surfaces as quotients of the upper
complex half--plane $ \cH =\{ z=x+iy\,|\,y>0\}$, with
Poincar\'e metric $ds^2 = y^{-2}(dx^2 +dy^2 )$ of constant
Gaussian curvature $K=-1$, by discrete subgroups $\Ga$ of
$PSL(2,\rz)$, known as {\it Fuchsian groups} (of the first
kind). $\Ga$ operates on $\cH$ by fractional linear
transformations: for
$\ga =\left( a\ b \atop c\ d \right) \in \Ga$, $z\in
\cH$, one sets $\ga z =(az+b)(cz+d)^{-1}$. The quantum versions
of these systems are governed by the Hamiltonian $H=-\De$,
where $\De =y^2 (\partial^2_x +\partial^2_y )$ is the hyperbolic
Laplacian ($\hbar =1=2m$). $H$ is an operator on the Hilbert
space $L^2 (\Ga \backslash \cH )$ of square integrable
functions that are invariant under the $\Ga$--operation
on $\cH$, i.e. $\psi (\ga z)=\psi (z)$ for all $\ga \in \Ga$.
For a certain subclass of these systems, characterized by
number theoretical properties of the Fuchsian group,
the notion of {\it arithmetical chaos} has been introduced
recently \cite{ArithmChaos} and it has been noted that the
energy level statistics of these systems violate the universal
laws of RMT. Several peculiarities appearing in arithmetical
chaos have been worked out, e.g. the presence of an infinite
algebra of hermitian operators commuting with the Hamiltonian
(the {\it Hecke operators}) and an exponential growth of the
multiplicities of lengths of classical periodic orbits.
Since Gutzwiller invented his by now famous trace formula
\cite{Gutz}, {\it periodic orbit theory} (POT) has been
developed into a major and powerful tool to investigate
the semiclassical quantization of chaotic dynamical systems.
POT connects the energy spectrum of a quantum system to the set
of periodic orbits of the underlying classical system.
Because of this relationship one could hope to trace back the
peculiarities of the energy spectra in arithmetical chaos
to properties of the classical periodic orbits. For the geodesic
flows on hyperbolic surfaces POT is even exact and not only a
semiclassical approximation. The trace formula was known as
the {\it Selberg trace formula} \cite{Sel} in mathematics long
before Gutzwiller treated the more general physical systems.
It is the geodesic length spectrum that in these cases exactly
determines the eigenvalues of the Laplacian. These facts may serve
as a motivation to study length spectra of periodic orbits
in arithmetical chaos more thoroughly.
The aim of this article now is to first explain the exact definition
of arithmetical chaos in more detail and then to prove the
law for the growth of the multiplicities of lengths of
periodic orbits, a result which has been announced before
independently in the two papers of ref. \cite{ArithmChaos}. In addition
we are now able to derive the constant multiplying the exponential,
a problem which was left open in \cite{ArithmChaos}.
This paper is organized in the following way. The next section
will contain a discussion of geodesic length spectra of
general hyperbolic surfaces. It will introduce the necessary
notation for subsequent considerations and relate length
spectra of surfaces whose Fuchsian groups are commensurable.
A collection of relevant definitions and facts from
algebraic number theory will be presentd in section 3. These
are needed to formulate the definition of
arithmetic Fuchsian groups. The next section then
will be devoted to a detailed discussion of length spectra
that occur on hyperbolic surfaces with arithmetic Fuchsian
groups. Out of this the main result on the mean
multiplicities of geodesic lengths in the case of arithmetic
groups will grow. This statement will be followed
by a presentation of two explicit examples for which
our result will be checked. A final discussion will then close
the present article.
%
\section{Length Spectra of Hyperbolic Surfaces}
Since according to the Selberg trace formula \cite{Sel}
the spectrum of the Laplacian on a hyperbolic surface
$M=\Ga \backslash \cH$ is
determined by its geodesic length spectrum,
we are interested in the geometry of such surfaces
when $\Ga$ is a Fuchsian group
of the first kind, i.e. a discrete subgroup of $PSL(2,\rz)$
such that $M$ has finite hyperbolic area.
In some loose notation the elements $\ga
\in \Ga$ will also be viewed as matrices in $SL(2,\rz)$ and
the identification of the matrices $\ga$ and $-\ga$ in
$PSL(2,\rz)=SL(2,\rz)/\{\pm \unmat \}$ will be understood
automatically. Thus $tr\,\ga$ denotes the corresponding
matrix--trace and we agree to choose it always
non--negative.
The fundamental group of $M$ is isomorphic to $\Ga$, where
the (free) homotopy classes in the fundamental group correspond
to conjugacy classes of hyperbolic elements in $\Ga$.
($\ga \in \Ga$ is called hyperbolic if $tr\,\ga >2$.) Each
hyperbolic conjugacy class in $\Ga$ represents a closed geodesic
on $M$ and its length $l(\ga )$, where $\ga$ is a representative
from the appropriate conjugacy class $\{\ga\}_{\Ga}$ in $\Ga$,
is related to the trace of $\ga$ by
\beq \label{length}
tr\,\ga =2\cosh \left( \frac{l(\ga )}{2} \right) \ .
\eeq
The determination of the geodesic length spectrum thus is equivalent
to describing the set of traces of inconjugate hyperbolic
elements in the Fuchsian group $\Ga$. In section 4 we therefore
will study which traces of group elements will occur for
arithmetic Fuchsian groups.
This section, however, will be devoted to a detailed investigation
of length spectra of general hyperbolic surfaces to
prepare for the more specific later considerations. Such a length
spectrum may be degenerate, i.e. there possibly exist
distinct geodesics on the surface sharing the same length $l$. In
this case the multiplicity of $l$ will be denoted by $g(l) \in \nz$.
In addition we
will call a geodesic {\it primitive} if it is traversed
only once. Then a corresponding representative $\ga \in \Ga$
is also primitive, i.e. it is not a power ($\geq 2$)
of some other element in $\Ga$. The primitive geodesics give
rise to the primitive length spectrum, which is the object we are
primarily interested in. The knowledge
of the sets of distinct lengths $\cL (\Ga )$ and of
distinct primitive lengths $\cL_p (\Ga )$
then allows to investigate the mean multiplicities
$$ and $$, respectively. This is possible since
for all Fuchsian groups {\it Huber's law} \cite{Huber}
universally determines the proliferation of geodesic lengths. It
states, using the notations $\cL (\Ga )=\{ l_1 $ of the multiplicities of primitive lengths
as in \cite{Aurich1}.
In section 4 we mainly will not deal with the primitive but
rather with the complete length spectrum. But we observe that
\beq \label{Nl}
\hat N(l)=\sum_{l_n \leq l}1 = \sum_{r=1}^{[l/l_1 ]}\sum_{rl_{p,n}
\leq l}1 =\sum_{r=1}^{[l/l_1 ]}\hat N_p (l/r)\ .
\eeq
Since $\hat N_p (l)$ is positive and monotonically increasing,
asymptotically for $l\rto \infty$ the sum on the very right of
(\ref{Nl}) is dominated by the first term $r=1$. Thus $\hat N(l)
\sim \hat N_p (l)$ for $l\rto \infty$. We conclude that in order
to gain information on the asymptotics of $$ for
$l\rto \infty$ one has to count the distinct geodesic lengths
up to $l$ in the limit $l\rto \infty$. Performing this for
arithmetic Fuchsian groups will be the main task in section 4.
It is known that for general hyperbolic surfaces $g_p (l)$ is
always unbounded \cite{Randol}. A crude estimate of how
frequently high values for $g_p (l)$ might occur shows
that these will show up very scarcely. Numerical computations
of the lower parts of length spectra for several arbitrarily
chosen compact surfaces of genus two show \cite{Aurich2} that
$g_p (l)$ never exceeds four in the computed range, a value
expected by symmetry arguments.
One might therefore speculate that high values for the
multiplicities of primitive lengths are so much suppressed
that they do not influence the asymptotics of their mean
$$. The latter rather seems to approach a constant value
determined only by the order of the group of geometric symmetries
(isometries) of $M$. The situation, however, changes drastically
if $\Ga$ is an arithmetic group. For $\Ga =PSL(2,\gz )$
e.g. it has been found that $\sim 2\,e^{l/2}/l$,
$l\rto \infty$. Another example of an arithmetic group that
has been treated before is that of the {\it regular octagon
group}. In \cite{Bogo} it is shown that $\sim 8\sqrt{2}\,
e^{l/2}/l$, $l\rto \infty$, for this surface.
The observation appearing in
\cite{ArithmChaos} now states that for every arithmetic Fuchsian
group $\sim const.\,e^{l/2}/l$, $l\rto \infty$,
is realized. This is significantly different from what is
found in the ``generic'' cases mentioned before.
For reasons that will become clear in section 4, we are interested
in the relation of length spectra for commensurable Fuchsian
groups. (We remark that two subgroups $H_1$ and $H_2$ of a
group $G$ are said to be {\it commensurable} if their intersection
is a subgroup of finite index in both $H_1$ and $H_2$.)
But let us first consider the case of a Fuchsian group $\Ga_1$
that is a subgroup of finite index $d$ in some other Fuchsian
group (of the first kind) $\Ga_2$. The generalization of the
result on the relation of the counting
functions $\hat N (l)$, to be achieved in the subsequent
reasoning, to the case of commensurable groups will then
easily follow, see (\ref{equiv})--(\ref{commen}).
The first step will be to notice that if $\ga \in \Ga_2$ then
there exists a $k\in \nz$ such that $\ga^k \in \Ga_1$.
To see this take any $\ga \in \Ga_2$ and form $\cup_{m\in \gz}\
\Ga_1 \ga^m \subset \Ga_2$. The union cannot be disjoint since
$\Ga_1$ is of finite index in $\Ga_2$. Therefore there exists
a $\ga_0 \in \Ga_1 \ga^r \cap \Ga_1 \ga^s $ for some pair
$r \neq s$. Thus $\ga^{r-s}$ and $\ga^{s-r}$ lie in $\Ga_1$.
Choosing $k=|r-s|$ then proves the assertion.
The set of all distinct primitive lengths derived from $\Ga_2$
will be denoted by
\beq \label{LS2}
\cL_p (\Ga_2 ) =\{ l_{p,1}From this observation one draws the main result of this section, namely
\beq \label{commen}
\hat N^{(a)}_p (l) \sim (d_a / d_b)\,\cdot\,\hat N^{(b)}_p (l)\ , \ \ \
l\rto \infty \ .
\eeq
Therefore, given two commensurable Fuchsian groups, their numbers
of distinct (primitive) lengths up to a given value $l$ are, in
the limit $l\rto \infty$, proportional to one another. The factor
of proportionality is given by the ratio of the indices with
which the two groups contain their intersection as a subgroup.
This nice result will find an application in section 4.
%
\section{Arithmetic Fuchsian Groups}
In this section we want to recall the definition of
{\it arithmetic Fuchsian groups} that give rise to the
chaotic systems showing arithmetical chaos.
They are Fuchsian groups of the first kind that
have special properties due to their number theoretical nature.
In general it will not be possible to define an arbitrary Fuchsian
group by giving rules that determine the matrix entries of
the group elements explicitly. Usually, $\Ga$ will be
characterized by some explicit generator matrices, which in
many cases are constrained by one or several relations. One of
the most prominent examples of a Fuchsian group is the modular
group $PSL(2,\gz)$. This, however, can easily be characterized
since it consists of all $2\times 2$--matrices of unit
determinant with integer entries. These matrix entries are
therefore given by a simple arithmetic statement. The most
general possible extension of this example now is formed by
the arithmetic Fuchsian groups. Their group elements are
given by matrices with certain entries from an
algebraic number field. The precise definition of these
groups requires some algebraic number theory, and for the
convenience of the reader, this section contains a short survey
of those aspects of algebraic number theory that are relevant
for the subsequent discussion. For a further reference, see
\cite{ANT,Vigneras,Shimura}.
An extension $K$ of finite degree $n$ of the field of rational
numbers $\Q$ is a field that contains $\Q$ as a subfield and,
viewed as a vector space over $\Q$ (in the obvious manner),
is of finite dimension $n$. Let $\Q [x]$ denote the ring of
polynomials in a variable $x$ with rational coefficients. $\al
\in K$ will be called {\it algebraic}, if it is a zero of some
polynomial from $\Q [x]$. The {\it minimal polynomial} of $\al$
is the (unique) element in $\Q [x]$ of lowest degree, and with
leading coefficient one, that has $\al$ as a root. The field $K$
is called an {\it algebraic number field}, if every $\al \in K$
is algebraic. Every extension $K$ of $\Q$ of finite degree is
known to be algebraic.
If $M$ is some arbitrary subset of $K$, $\Q (M)$ is defined to be
the smallest subfield of $K$ that contains both $M$ and $\Q$.
It is given by all values of all polynomials in the elements of
$M$ with rational coefficients, and all possible quotients thereof.
$\Q (M)$ is called the {\it adjunction} of $M$ to $\Q$. One
can now show that every algebraic number field $K$ of finite degree
over $\Q$ can be realized as an adjunction of a single algebraic
number $\al \in K$ to $\Q$; therefore $K=\Q (\al )$.
Since $K$ is a vector space of dimension $n$ over $\Q$, the $n+1$
algebraic numbers $1,\al,\dots ,\al^n $ have to be linearly
dependent and thus to obey a relation
\beq
a_n \al^n +\dots +a_1 \al +a_0 =0
\eeq
with rational coefficients $a_i$ and $a_n \neq 0$. Normalizing
the leading coefficient to one leaves $\al$ as a root of
an irreducible polynomial $f_{\al}(x)\in \Q [x]$ of degree $n$.
$f_{\al}(x)$ is the minimal polynomial of $\al$. Since $\{ 1,\al ,
\dots ,\al^{n-1} \}$ may serve as a basis for $K$ over $\Q$,
any $x\in K$ may be expanded as a linear combination of powers
of $\al$ up to the order $n-1$,
\beq
x=b_{n-1}\al^{n-1}+\dots +b_1 \al +b_0 \ ,
\eeq
with rational coefficients $b_i$.
The polynomial $f_{\al}(x)$ has $n$ different complex roots $\al_1 ,
\dots ,\al_n$ $(\al_1 =\al )$. One can thus define $n$ different
homomorphisms
$\vp_i :\ K\lto \C$, $i=1,\dots ,n$, that leave $\Q$ invariant, by
\beq
\vp_i (x):= b_{n-1}\al_i^{n-1}+\dots +b_1 \al_i +b_0 \ ,
\eeq
$\vp_1 (x)=x$. The $\vp_i$'s are called the {\it conjugations} of
$K$. If all images of $K$ under these homomorphisms
are contained in the real numbers, $K$ is said to be
{\it totally real}.
On $\Q$ the usual absolute value $\nu_1 (x)=|x|$, $x\in \Q$,
introduces a topology, which is, however, not complete.
The $n$ conjugations $\vp_i $ offer $n$ distinct ways to
embed $K$ into $\rz$. Thus $n$ different absolute values
$\nu_i$ are given on $K$ by $\nu_i (x):= |\vp_i (x)|$,
and these can be used to complete $K$
to $K_{\nu_i}\cong \rz$. The $\nu_i$'s are also called the
(archimedean) {\it infinite primes} of $K$.
All algebraic numbers in $K$ whose minimal polynomials have
coefficients in the rational integers $\gz$ form a
ring $\cR_K $, which is called the {\it ring of integers of} $K$.
An element $x\in \cR_K $ is also called an {\it algebraic integer}.
A $\gz$--module (i.e. an additive abelian group)
$o\subset K$ of (the maximal possible) rank $n$ that at
the same time is a subring of $K$ is called an {\it order} of $K$.
Since we understand a ring to contain a unity, every order $o
\subset K$ contains the rational integers $\gz$. Further it is
known that there exists a {\it maximal order} in $K$ that
contains all other orders, and that this maximal order is just the
ring of algebraic integers $\cR_K$. An order $o$ possesses a
module--basis of $n$ algebraic numbers $\om_1 ,\dots ,\om_n $
that are linearly independent over $\gz$ and hence also,
equivalently, over $\Q$,
\beq
o= \gz \om_1 \oplus \dots \oplus \gz \om_n \ .
\eeq
The {\it discriminant} of $o$ is defined to be $D_{K/ \qz}(o):=
[det (\vp_j (\om_i ))]^2 \neq 0$. In complete analogy one can
also define a discriminant for any $\gz$--module of rank $n$ in $K$.
Another important notion, to be introduced now, is that of a
{\it quaternion algebra}. In doing so, we will mainly follow
\cite{Vigneras,Shimura}.
An algebra $A$ over a field $K$ is
called {\it central}, if $K$ is its center; it is said to be
{\it simple}, if it contains no two--sided ideals besides
$\{ 0\}$ and $A$ itself. A quaternion algebra then is defined
to be a central simple algebra $A$ of dimension four over $K$.
In more explicit terms $A$ may be visualized as follows:
the elements of a basis $\{ 1, \al ,\be ,\ga \}$ of $A$ over
$K$ have to obey the relations $\ga =\al \be =-\be \al$,
$\al^2 =a$, $\be^2 =b$; $a,b \in K\backslash \{0\}$.
Any $X\in A$ may then be expanded as
\beq \label{basis}
X=x_0 +x_1 \al +x_2 \be +x_3 \ga \ ,
\eeq
with $x_0 ,\dots , x_3 \in K$. On $A$ there exists an involutory
anti--automorphism, called the {\it conjugation} of $A$, that
maps $X$ to $\bar X:= x_0 -x_1 \al -x_2 \be -x_3 \ga$. Thus
$\overline{\bar{X}}=X$ and $\overline{X\cdot Y}=\bar{Y}\cdot \bar{X}$.
The conjugation enables one to define the {\it reduced trace}
and the {\it reduced norm} of $A$,
\beqa
tr_A (X) &:=& X+\bar{X}=2x_0 \ , \nonumber \\
n_A (X) &:=& X\cdot \bar{X}= x_0^2 -x_1^2 a-x_2^2 b +x_3^2 ab \ .
\eeqa
If $A$ is a division algebra, i.e. if every $X\neq 0$ in $A$
possesses an inverse, $n_A (X) =0$ implies $X=0$. The
inverse is then given by $X^{-1}=\frac{1}{n_A (X)} \bar{X}$.
A $\gz$--module $\cO \subset A$ of (the maximal possible) rank $4n$
that also is a subalgebra in $A$ is called an {\it order} of $A$.
The introduction of a module--basis $\{ \tau_1 ,\dots ,
\tau_{4n} \}$ turns the order into
\beq \label{order}
\cO =\gz \tau_1 \oplus \dots \oplus \gz \tau_{4n} \ .
\eeq
We further introduce the {\it group of units of norm one}
$\cO^1 := \{ \ve \in \cO \,|\, \ve^{-1}\in \cO ,\ n_A (\ve)=1\,\}$.
A well--known example of a (division) quaternion algebra
is given by {\it Hamilton's quaternions}
$$
\H := \left\{ \left( \begin{array}{cc} z & w \\ -\bar w & \bar z
\end{array} \right) \ |\ z,w \in \C \right\}\ .
$$
$\H $ is a four dimensional $\rz$--subalgebra of $M(2,\C )$,
the algebra of complex $2\times 2$-- matrices, characterized
by the parameters $a=b=-1$. The subgroup of elements of
reduced norm one is just $SU(2,\C )$. An even simpler example
of a (non--division) quaternion algebra over $\rz$ is
$M(2,\rz )$. In fact, $\H$ and $M(2,\rz )$ are the only
quaternion algebras over $\rz$.
A classification of quaternion algebras over $K$ can now be
achieved by looking at the corresponding algebras over $\rz$
with the help of the $n$ completions $K_{\nu_i}\cong \rz$.
Define $A_i := A \otimes_{\qz} K_{\nu_i} \cong A \otimes_{\qz}
\rz$, which is a quaternion algebra over $\rz$. Hence it is either
isomorphic to $\H$ (if it is a division algebra), or to
$M(2,\rz )$ (if it is a non--division algebra). For the definition
of arithmetic Fuchsian groups (see \cite{Vigneras,Takeuchi})
we consider the case $A_1 \cong
M(2,\rz )$ and $A_i \cong \H$ for $i=2,\dots ,n$.
Therefore there exists an isomorphism
\beq \label{Iso}
\rho :\ \ A\otimes_{\qz} \rz \lto M(2,\rz )\oplus \H \oplus \dots
\oplus \H \ ,
\eeq
where there occur $n-1$ summands of $\H$. $\rho_j$ will denote
the restriction of $\rho$ to $A$ followed by a projection onto
the $j$-th summand in (\ref{Iso}).
The several reduced traces and norms for $X\in A$ in (\ref{Iso})
are related by
\beqa \label{det}
tr\,\rho_1 (X) &=& tr_A (X)\ , \nonumber \\
det\,\rho_1 (X) &=& n_A (X) \ , \nonumber \\
tr_{\H}\rho_j (X) &=& \vp_j (tr_A (X))=\vp_j (tr\,\rho_1 (X))\ ,
\nonumber \\
n_{\H}(\rho_j (X))&=& \vp_j (n_A (X))=\vp_j (det\,\rho_1 (X))\ ,
\ \ \ \ j=2, \dots ,n\ .
\eeqa
The image of $A$ under $\rho_1$ in $M(2,\rz )$ may also be
expressed in more explicit terms by using the basis $\{1,\al ,
\be ,\al \be \}$ for $A$, see (\ref{basis}): $\rho_1 (1)$ is the
$2\times 2$ unit matrix; $\rho_1 (\al )$ and $\rho_1 (\be )$
may be represented, by using the parameters $a,b >0$, as
\beq
\rho_1 (\al )=\left( \begin{array}{cc} \sqrt{a} & 0 \\ 0 &
-\sqrt{a} \end{array} \right)\ , \ \ \ \ \
\rho_1 (\be )=\left( \begin{array}{cc} 0 & \sqrt{b} \\
\sqrt{b} & 0 \end{array} \right)\ .
\eeq
For $X=x_0 +x_1 \al + x_2 \be + x_3 \al \be \in A$ the matrix
$\rho_1 (X)$ in this representation takes the form
\beq \label{matrix}
\rho_1 (X)=\left( \begin{array}{cc} x_0 +x_1 \sqrt{a} & x_2 \sqrt{b}
+x_3 \sqrt{ab} \\ x_2 \sqrt{b}-x_3 \sqrt{ab} & x_0 -x_1 \sqrt{a}
\end{array} \right) \ .
\eeq
We are now seeking for a subset in $A$ whose image under $\rho_1$
in $M(2,\rz)$ gives a Fuchsian group $\Ga$. Therefore
$\rho_1^{-1} (\Ga)$ must be a discrete multiplicative subgroup
of $A$. Furthermore, for $\rho_1 (X)=\ga\in \Ga$ the
condition $det\,\ga =1$ must be fulfilled. Thus by (\ref{det})
$n_A (X)=1$ has to be required. Hence we are led to look
at groups of units of norm one $\cO^1$ of orders $\cO \subset
A$. Regarding their images under $\rho_1$ one finds in
\cite{Shimura,Takeuchi} the following \\
\noindent {\sc Proposition:} Let $A$ be a quaternion algebra
over the totally real algebraic number field $K$ of degree
$n$. Let $\cO \subset A$ be an order and $\cO^1$ be its
group of units of norm one. Then $\Ga (A,\cO ):=\rho_1 (\cO^1 )$
is a Fuchsian group of the first kind. Moreover, $\Ga (A,\cO )
\backslash \cH$ is compact if $A$ is a division algebra. A change
of the isomorphism $\rho$ in (\ref{Iso}) amounts to a conjugation of
$\Ga (A,\cO )$ in $SL(2,\rz )$. \\
The proposition now tells us that we have found what we were
looking for: a class of arithmetically defined Fuchsian groups.
We are aiming at counting the numbers of distinct primitive
lengths to gain information on the mean multiplicities in the
length spectra derived from the Fuchsian groups under
consideration. For this purpose (\ref{commen}) allows
to enlarge the class of groups appearing in the proposition
a little. \\
\noindent {\sc Definition:} A Fuchsian group $\Ga$ that is a
subgroup of finite index in some $\Ga (A,\cO )$ will be
called a {\it Fuchsian group derived from the quaternion algebra
A}. (The shorthand phrase {\it quaternion group} will also be
sometimes used instead.) A Fuchsian group $\Ga$ that is
commensurable with some $\Ga (A,\cO )$ will be called an
{\it arithmetic Fuchsian group}. \\
Finally, we mention that in \cite{Takeuchi} a characterization of
arithmetic Fuchsian groups $\Ga$ is presented in terms of an
adjunction of $tr\,\Ga =\{tr\,\ga \,|\,\ga \in \Ga \}$ to
$\Q$. This already shows that the traces of elements of arithmetic
groups (and hence also the lengths of closed geodesics on
$\Ga \backslash \cH$) share special features that distinguish
them from the non--arithmetic case.
%
\section{Counting Traces in Arithmetic Groups}
After having discussed length spectra of hyperbolic surfaces
and having introduced the concept of arithmetic Fuchsian groups,
we want to investigate the length spectra, i.e.
the sets of traces occurring in these
groups. As in the preceding section, take $A$ to be a
quaternion algebra over a totally real algebraic number
field $K$ of degree $n$ and $\cO$ as some order in $A$. By
the proposition from section 4 $\Ga :=\Ga (A,\cO ) =\rho_1
(\cO^1 )$ is a Fuchsian group of the first kind. For $X= x_0 +
x_1 \al +x_2 \be +x_3 \al\be \in \cO$ denote $\rho_1 (X)=
\ga \in \Ga$. By (\ref{matrix}) one sees that $\frac{1}{2}tr\,\ga
=x_0 \in \cO |_K =:\cM$. Thus
\beq
tr\,\Ga =tr_A \cO^1 \subset tr_A \cO = 2\cM \ .
\eeq
The inclusion $ tr_A \cO^1 \subset tr_A \cO $ will in general
be a proper one and we will return to this problem later.
The aim now is to determine the number $\hat N_p (l)$
of distinct primitive lengths on $\Ga \backslash \cH$ for
$l\rto \infty$. By (\ref{length}) one hence has to count
the number of distinct traces in $\Ga$ with $20$, $l\rto \infty$.
We are now in a position to state our main result as the \\
\noindent {\sc Theorem:} Let $\Ga$ be an arithmetic Fuchsian
group, commensurable with the group $\Ga (A, \cO )$ derived
from the quaternion algebra $A$ over the totally real
algebraic number field $K$ of degree $n$.
Denote by $d_1$ the index of the subgroup $\Ga_0 := \Ga \cap
\Ga (A,\cO )$ in $\Ga$, and by $d_2$ the respective index of
$\Ga_0$ in $\Ga (A,\cO )$. Let $D_{K/\qz}(\cM)$ be the
discriminant of the module $\cM \subset K$ that contains
$\frac{1}{2}tr\,\Ga (A,\cO )$.
Then, under the hypothesis (\ref{Hyp}), the number $\hat N_p (l)$
of distinct primitive lengths on $\Ga \backslash \cH$ up
to $l$ grows asymptotically like
\beq \label{Theorem}
\hat N_p (l)\sim 2^{n-2}\,\cdot\,(d_1 /d_2)\,\cdot\,[D_{K/\qz} (\cM)]^{
-1/2} \,\cdot\ e^{l/2}\ ,\ \ \ l\rto \infty\ .
\eeq
\noindent {\sc Proof:} Assume the validity of the hypothesis
(\ref{Hyp}) and recall the asymptotic relation $\hat N_p (l)
\sim \hat N(l)$, $l\rto \infty$, from section 2. Therefore
also $\hat N_p (l) \sim \cN (\cosh (l/2))$. Using (\ref{commen})
and (\ref{35}) then leads to the assertion. \\
In \cite{Aurich1} it was shown how to derive the asymptotics
for $l\rto \infty$ of the local average $$ of the
multiplicities in the primitive length spectrum from that
of $\hat N_p (l)$ for cases like (\ref{Theorem}). We thus
observe the \\
\noindent {\sc Corollary:} The local average of the primitive
multiplicities in the cases described in the theorem
behaves asymptotically like
\beq \label{Corollary}
\sim 2^{3-n}\ \frac{d_2}{d_1}\ \sqrt{D_{K/\qz} (\cM)}\ \cdot\,
\frac{e^{l/2}}{l}\ ,\ \ \ l\rto \infty\ .
\eeq
%
\section{Two Examples of Arithmetic Fuchsian Groups}
In this section we would like to discuss two examples of
arithmetic Fuchsian groups for which the asymptotic behaviour of
$$ was known before.
The first example will be the prototype one, namely the modular
group $\Ga =PSL(2,\gz )$. In this case the relevant number field
$K$ is just the field of rational numbers $\Q$ itself, which
obviously has degree $n=1$. There is only one order in $\Q$,
the maximal one $\cR_K =\gz$ of rational integers. The
quaternion algebra $A$ is characterized by the two parameters
$a=b=1$; thus $A=M(2,\Q )$. (We remark that $A$ is a non--division
algebra in accordance with the non--compactness of $\Ga \backslash
\cH$.) The order $\cO \subset A$ is determined by the four
elements
\beq \label{taubasis1}
\tau_1 =\left( \begin{array}{cc} 1&0\\0&0 \end{array}\right) \ ,\ \ \
\tau_2 =\left( \begin{array}{cc} 0&1\\0&0 \end{array}\right) \ ,\ \ \
\tau_3 =\left( \begin{array}{cc} 0&0\\1&0 \end{array}\right) \ ,\ \ \
\tau_4 =\left( \begin{array}{cc} 0&0\\0&1 \end{array}\right)
\eeq
of the $\gz$--basis for $\cO$. By (\ref{order}) therefore
$\cO =M(2,\gz )$, which obviously leads to $\cO^1 =SL(2,\gz )$.
Expand $X\in \cO$ into the basis (\ref{taubasis1}), $X=k_1 \tau_1
+\dots + k_4 \tau_4 $, $k_i \in \gz$, from which one observes that
$\frac{1}{2}tr_A X =x_0 =\frac{1}{2}(k_1 +k_4 )$. This yields
$\cM =\frac{1}{2}\gz$ and $\mu_1 =\frac{1}{2}$, see (\ref{M}).
It is known \cite{Schleicher} that for the modular group $tr\,\Ga
=\gz =2\cM$ and therefore no gaps in the set of traces occur.
The discriminant of $\cM$ now is trivially obtained, and
$\sqrt{D_{\qz}(\cM)}=\mu_1 =\frac{1}{2}$. As the modular group
itself is the group $\cO^1 =\Ga (A,\cO )$ just defined, one
concludes using $d_1 =d_2 =1$,
\beqa
\hat N_p (l)&\sim& e^{l/2} \ , \nonumber \\
&\sim& \frac{e^{l/2}}{l/2}\ ,\ \ \ l\rto \infty \ ,
\eeqa
which agrees with the previously known result.
As a second example we would like to introduce the regular octagon
group $\Ga_{reg}$, see \cite{Aurich1,Bogo}. This is a Fuchsian group
that leads to a compact surface $\Ga_{reg}\backslash \cH$ of genus
two, which is the most symmetric one of this type. $\Ga_{reg}$
is a normal subgroup of index 48 in another Fuchsian group.
Including orientation--reversing diffeomorphisms the group
of isometries of this surface thus is of order 96. As an
arithmetic group $\Ga_{reg}$ may be obtained in the
following way. The algebraic number field $K=\Q (\sqrt{2})$
to be considered is of degree $n=2$. The ring of
integers in it is $\cR_K =\gz [\sqrt{2}] =\{m+n\sqrt{2}\,|\, m,n
\in \gz \}$. A basis $\{ \om_1 ,\om_2 \}$ for $K$ which at the
same time is a module--basis for $\cR_K$ is given by
$\{ 1,\sqrt{2} \}$. The two parameters determining the
quaternion algebra $A$ are $a=1+\sqrt{2}$ and $b=1$.
We now characterize the relevant order $\cO$ in $A$ by
specifying the module--basis $\{ \tau_1 ,\dots , \tau_8 \}$
for $\cO$ as $\{ \om_1 \cdot 1,\dots ,\om_2 \cdot \al\be \}$.
Therefore, by (\ref{order}), an element $\ga =\rho_1 (X)$ for
$X\in \cO$ can be identified as
\beq \label{groupmatrix}
\ga =\left( \begin{array}{cc} x_0 + x_1 \sqrt{1+\sqrt{2}} &
x_2 + x_3 \sqrt{1+\sqrt{2}} \\ x_2 - x_3 \sqrt{1+\sqrt{2}} &
x_0 - x_1 \sqrt{1+\sqrt{2}} \end{array} \right) \ ,
\eeq
with $x_i =m_i +n_i \sqrt{2}$, $m_i ,n_i \in \gz$. From
this one concludes that $x_0 \in \cM =\gz [\sqrt{2}]$.
In this (exceptional) case therefore $\cM$ is an order in
$K$; it even is the maximal one. A basis for $\cM$ then
is given by $\{ \mu_1 ,\mu_2 \} =\{ 1,\sqrt{2} \}$ and thus
${\bf e}_1 =(1,1)$ and ${\bf e}_2 =( \sqrt{2}, -\sqrt{2} )$.
This allows to determine the discriminant of $\cM$, leading
to $\sqrt{D_{K/\qz}(\cM )}=| det(e_{ij}) | = 2\sqrt{2}$.
The group $\Ga (A,\cO )$ now is formed by all matrices
(\ref{groupmatrix}) of unit determinant. In \cite{Pignataro}
it is shown that this group may also be obtained by
adjoining an additional element $S=\left( \ 0\,1 \atop -1\,0 \right) $
to $\Ga_{reg}$. Therefore, $\Ga (A,\cO )=\Ga_{reg} \dcup
\Ga_{reg} S$, and $\Ga_{reg}$ is a subgroup of
index two in $\Ga (A,\cO )$. From the theorem one hence concludes
(using $d_1 =1$, $d_2 =2$)
\beqa
\hat N_p (l) &\sim& \frac{1}{4\sqrt{2}}\,\cdot \ e^{l/2} \ ,
\nonumber \\
< g_p (l)> &\sim& 8\sqrt{2}\,\cdot \ \frac{e^{l/2}}{l}\ ,
\ \ \ l\rto \infty \ .
\eeqa
This is exactly the result found in \cite{Aurich1,Bogo}.
We remark that in this example there do exist gaps in the
length spectrum. These have been identified in \cite{Aurich1,Bogo},
where it was further shown by a numerical calculation
of the geodesic length spectrum up to $l=18$ that these
do not alter the result and that therefore
our hypothesis (\ref{Hyp}) is fulfilled.
%
\section{Discussion}
This article contained a study of geodesic length spectra on
hyperbolic surfaces $\Ga \backslash \cH$, where $\Ga$
is a Fuchsian group of the first kind. In the first part
of this investigation we discussed how the counting
functions for distinct lengths and for distinct primitive
lengths, respectively, on such a surface are asymptotically related.
Moreover, we looked at relations among the counting
functions for two different surfaces whose Fuchsian groups
are commensurable. It was found that these share essentially
the same asymptotic behaviour (besides explicitly
known factors of proportionality).
After this discussion we continued in explaining the
definition of arithmetic Fuchsian groups and provided
definitions and properties of the relevant quantities
from number theory. In doing so we tried to be as explicit
as possible and to avoid unnecessary generality.
We were then able to count the number of different geodesic
lengths asymptotically up to a given (large) value. This
was possible, since we could identify the set of traces
that occurs for an arithmetic group as being (the essential
part of) a module in the field of algebraic numbers used
in the definition of the group. We were able to map
this module to a lattice in $\rz^n $. The counting problem
was thus reformulated as a lattice point problem.
It was found that the number of distinct lengths up to a
value of $l$ grows like $e^{l/2}$ for $l\rto \infty$,
whereas by Huber's law the number of geodesics with lengths
up to $l$ grows like $e^l \,/l $, which is exceptionally
strong compared to the case of non--arithmetic groups.
The reason for this difference --the number of different
lengths proliferates only like, roughly speaking, the
square root of the number of geodesics-- is caused by
the arithmetic restriction on the set of possible lengths,
which does not exist for non--arithmetic groups. For
these, however, such a quite general description as we
presented it here in the arithmetic case seems not to be possible.
In proving our result we also gave a receipt to find the
complete asymptotic expression for the mean multiplicity
$$ of primitive lengths, including the overall constant.
First, given an arithmetic Fuchsian group $\Ga $,
one has to know the quaternion group $\Ga (A,\cO )$
commensurable with it. Once one has this at hand one also
knows the indices $d_1$ and $d_2$ describing $\Ga \cap \Ga (A,\cO )$
as a subgroup in $\Ga$ and in $\Ga (A,\cO )$. In \cite{Takeuchi}
it is shown that one can get the relevant number field $K$
by adjoining $tr\,\Ga (A,\cO )$ to $\Q$. Furthermore, the algebra
$A$ can be received as the linear span of $\Ga (A,\cO )$
over $K$. Analogously, the order $\cO \subset A$ is obtained
as the linear span of $\Ga (A,\cO )$ over $\cR_K$. One then has
to find the module--basis $\{ \tau_1 ,\dots ,\tau_{4n} \}$ of
$\cO$. This can be used to obtain the matrix $(M_{i,j})$
appearing in (\ref{basechange}). Given this one has to
identify the module $\cM$ containing $\frac{1}{2}tr\,\Ga (A,\cO )$
(see (\ref{littleo}) and (\ref{M})) in order to determine
its discriminant $D_{K/\qz}(\cM)$. One can now plug all this
information into (\ref{Corollary}) to get the answer
to the problem.
The arithmetic nature of the groups considered in this
article does also produce restrictions on the quantum
systems that arise as quantizations of the geodesic flows
on the corresponding surfaces. The already mentioned
existence of the infinitely many self--adjoint Hecke
operators commuting with the Hamiltonian leads to
correlations among the values in each wavefunction,
see the first paper of ref. \cite{ArithmChaos}.
This is in sharp contrast to the non--arithmetic case where
at most a finite number of such Hecke operators can exist
and thus the correlations in the wavefunctions are by far
weaker (if present at all). In \cite{ArithmChaos} it was
argued that this was the reason why the statistical properties
of the energy spectra in the arithmetical case are different
from ``generic'' chaotic systems. As the Selberg trace
formula connects the geodesic length spectrum to the spectrum
of the hyperbolic Laplacian, one could expect a link between
the exceptional status' of surfaces with arithmetic Fuchsian
groups in both the classical and quantum context. This,
however, has still to be made more explicit.
%
\section*{Acknowledgements}
I would like to thank Prof. F. Steiner for useful discussions
and for reading the manuscript. Further I want to thank
Holger Ninnemann for helpful discussions.
%
\begin{thebibliography}{99}{\footnotesize
\bibitem{Berry} M.V. Berry and M. Tabor, Proc. Roy. Soc. London
{\bf A356}(1977)375
\bibitem{Sinai} Ya.G. Sinai, {\it Mathematical problems in the
theory of quantum chaos}, Distinguished Raimond and Beverly
Sackler lectures, Tel Aviv University, 1990 (to appear in Springer
Lecture Notes in Mathematics) \\
P.M. Bleher, in {\it Mathematical Physics X}, Proc. of the
$X^{th}$ Congress on Mathematical Physics, Leipzig 1991, p. 298,
ed.: K. Schm\"udgen (Springer, Berlin--Heidelberg--New York, 1992)
\bibitem{RMT} M.L. Mehta, {\it Random Matrices and the Statistical
Theory of Energy Levels} (Academic Press, New York, 1967); new
revised and enlarged edition (Academic Press, New York, 1990) \\
O. Bohigas, in {\it Chaos and Quantum Physics}, Proc. of the
Les Houches Summer School 1989, eds.: M.-J. Giannoni, A. Voros
and J. Zinn-Justin (Elsevier Science Publ., Amsterdam, 1991) \\
F. Haake, {\it Quantum Signatures of Chaos} (Springer,
Berlin--Heidelberg, 1991)
\bibitem{GOE} S.W. McDonald and A.N. Kaufmann, Phys. Rev. Lett.
{\bf 42}(1979)1189 \\
M.V. Berry, Ann. Phys. {\bf 131}(1981)163 \\
O. Bohigas, M.-J. Giannoni and C. Schmit, Phys. Rev. Lett.
{\bf 52}(1984)1 \\
M.V. Berry and M. Robnik, J. Phys. {\bf A19}(1985)649\\
R. Aurich and F. Steiner, Physica {\bf D43}(1990)155 \\
M. Sieber and F. Steiner, Phys. Lett. {\bf A148}(1990)415
\bibitem{ArithmChaos} J. Bolte, G. Steil and F. Steiner,
{\it Arithmetical Chaos and Violation of Universality in Energy
Level Statistics}, DESY--preprint, DESY 92-061, April 1992 \\
E.B. Bogomolny, B. Georgeot, M.-J. Giannoni and C. Schmit,
{\it Arithmetical Chaos}, preprint IPNO/TH 92-23
\bibitem{Gutz} M.C. Gutzwiller, J. Math. Phys. {\bf 8}(1967)1979;
{\bf 10}(1969)1004; {\bf 11}(1970)1791;
{\bf 12}(1971)343 \\
M.C. Gutzwiller, {\it Chaos in Classical and Quantum
Mechanics} (Springer, New York, 1990)
\bibitem{Sel} A. Selberg, J. Indian Math. Soc. {\bf 20}(1956)47 \\
D.A. Hejhal, {\it The Selberg Trace Formula for $PSL(2,\rz)$, vol. I
and II}, Springer Lecture Notes in Mathematics {\bf 548}(1976)
and {\bf 1001}(1983) \\
A.B. Venkov, {\it Spectral Theory of Automorphic Functions and
its Applications} (Kluwer Academic Publ., Dordrecht--Boston--London,
1990)
\bibitem{Huber} H. Huber, Math. Annalen {\bf 138}(1959)1
\bibitem{Aurich1} R. Aurich and F. Steiner, Physica {\bf D32}(1988)451
\bibitem{Randol} R.D. Horowitz, Commun. Pure Appl. Math.
{\bf 25}(1972)635 \\
B. Randol, Proc. Amer. Math. Soc. {\bf 78}(1980)455
\bibitem{Aurich2} R. Aurich, private communication
\bibitem{Bogo} R. Aurich, E.B. Bogomolny and F. Steiner, Physica
{\bf D48}(1991)91
\bibitem{ANT} B.L. van der Waerden, {\it Algebra I}, 5. Auflage
(Springer, Berlin--G\"ottingen--Heidelberg, 1960) \\
S.I. Borewicz and I.R. \v{S}afarevi\v{c}, {\it Zahlentheorie}
(Birkh\"auser, Basel--Stuttgart, 1966)
\bibitem{Vigneras} M.-F. Vign\'eras, {\it Arithm\'etique des Alg\`ebres
de Quaternions}, Springer Lecture Notes in Mathematics {\bf 800}(1980)
\bibitem{Shimura} G. Shimura, {\it Introduction to the Arithmetic
Theory of Automorphic Functions} (Princeton University Press,
Princeton, 1971)
\bibitem{Takeuchi} K. Takeuchi, J. Fac. Sci. Univ. Tokyo
{\bf 16}(1969)97; J. Math. Soc. Japan {\bf 27}(1975)600
\bibitem{Lang} S. Lang, {\it Algebraic Numbers} (Addison--Wesley,
Reading--Palo Alto--London, 1964)
\bibitem{Schleicher} G. Handke, {\it \"Uber Streuzust\"ande in
einem klassisch chaotischen System} (Diploma Thesis, Univ. Hamburg,
1989) \\
D. Schleicher, {\it Bestimmung des Energiespektrums in einem
chaotischen System mit Hilfe der Selbergschen Spurformel} (Diploma
Thesis, Univ. Hamburg, 1991)
\bibitem{Pignataro} T. Pignataro, {\it Hausdorff Dimension, Spectral
Theory and Applications to the Quantization of Geodesic Flows
on Surfaces of Constant Negative Curvature} (PhD Thesis, Princeton
Univ., 1984)
}
\end{thebibliography}
\end{document}
ENDBODY