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% Author Package file for use with AMS-LaTeX 1.2
\controldates{22-JUN-1999,22-JUN-1999,22-JUN-1999,22-JUN-1999}
 
\documentclass{era-l}
\usepackage{graphicx}
\newtheorem{Th}{Theorem}

\newcommand\eps{\epsilon}
\newcommand\calF{\mathcal{F}}
\newcommand\calO{\mathcal{O}}
\newcommand\calU{\mathcal{U}}
\newcommand\R{\mathbb{R}}
\newcommand\syst{\operatorname{syst}}
\newcommand\diam{\operatorname{diam}}
\newcommand\genus{\operatorname{genus}}

\begin{document}

\title{The first eigenvalue of a Riemann surface}

\author{Robert Brooks}
\address{Department of Mathematics,
Technion---Israel Institute of Technology,
Haifa, Israel}
\email{rbrooks@tx.technion.ac.il}
\thanks{Partially supported by the Israel
Science Foundation, founded by the Israel Academy of Arts and
Sciences, the Fund for the Promotion of Research at the
Technion, and the New York Metropolitan Fund.}

\author{Eran Makover}
\address{Department of
Mathematics and Computer Science,
Drake University, 
Des Moines, IA 50311}
\curraddr{Department of Mathematics, Dartmouth College, Hanover, NH}
\email{eranm@math.huji.ac.il}

\issueinfo{5}{11}{}{1999}
\dateposted{June 28, 1999}
\pagespan{76}{81}
\PII{S 1079-6762(99)00064-5}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 58G99}

\date{March 25, 1999} 

\commby{Walter Neumann}

\begin{abstract}
We present a collection of results whose central theme is that
the phenomenon of the first eigenvalue of the Laplacian being large is
typical for Riemann surfaces. Our main analytic tool is a method for
studying how the hyperbolic metric on a Riemann surface behaves under
compactification of the surface. We make the notion of picking a Riemann
surface at random by modeling this process on the process of picking a
random $3$-regular graph. With this model, we show that there are positive
constants $C_1$ and $C_2$ independent of the genus, such that with
probability at least $C_1$, a randomly picked surface has first eigenvalue
at least $C_2$.
\end{abstract}
\maketitle


In this note, we announce a collection of results (\cite{BM1,BM2,BM3})
connected to the behavior of the first eigenvalue $\lambda_1(S)$ of a
compact Riemann surface of large genus, endowed with a metric of
constant curvature $-1$. These results have as their common theme that
the phenomenon of $\lambda_1$ large is in some sense typical. To make
the notion of ``typical'' precise, we model the process of picking a
Riemann surface at random on the process of picking a $3$-regular
graph at random. 

The idea of studying the first eigenvalue of a Riemann surface via the
study of eigenvalues of $3$-regular graphs comes from the work of
Buser \cite{Bu1,Bu2}. In effect, our approach here is a variation on
his idea, where we first study the behavior of $\lambda_1$ on
finite-area Riemann surfaces connected to $3$-regular graphs, and then
see how $\lambda_1$ changes when we compactify the surface.

Our main analytic tool is a method for studying how the hyperbolic
metric of a finite-area Riemann surface behaves under such a
compactification.  This method was introduced in \cite{PS}, and is based on
the Ahlfors-Schwarz Lemma (\cite{A}; see also \cite{GA}). 

We then have:

\begin{Th}[{\cite{BM1}}] \label{bma} 
For all $\eps$, there exists $N$ such that,
for $g \ge N$, there is a compact Riemann surface $S_g$ of genus $g$
satisfying 
\[\lambda_1(S_g) \ge {\frac{171}{784}} - \eps.\]
\end{Th}

The number $171/784$ comes from the improvement by Luo, Rudnick, and 
Sarnak
\cite{LRS} of the Selberg $3/16$ Theorem \cite{Sel}. If Selberg's
conjecture were true, we would be able to replace $3/16$ by $1/4$, the
best possible value. More generally, Theorem \ref{bma} contains a
method to build from a collection of Riemann surfaces $\{S_i\}$ of
large first eigenvalue a larger collection of surfaces whose genera
may include all but finitely many genera, whose first eigenvalues
satisfy a similar bound.

To state our next results, let $\calF_{n,k}$ denote the set of
$k$-regular graphs on $n$ vertices, and $\calF_{n,k}^*$ the set of pairs
$(\Gamma, \calO)$, where $\Gamma \in \calF_{n,k}$ and $\calO$ is an
orientation on $\Gamma$ ---that is, for each vertex $v$ of $\Gamma$,
$\calO$ prescribes a cyclic ordering of the edges emanating from
$\calO$. Let $\calF_{n,k}^!$ denote the subset of $\calF_{n,k}$
consisting of graphs without loops, double edges, or $3$-cycles.
As a probability
space, $\calF_{n,k}^!$ has positive measure in $\calF_{n,k}$ bounded
away from $0$ as $n \to \infty$.

We describe a way of associating to the pair $(\Gamma,\calO)\
\in \ \calF_{n,3}$ a compact Riemann surface $S^C(\Gamma, \calO)$. By a
theorem of Belyi \cite{Be}, the surfaces that arise in this way are
dense in the space of all compact Riemann surfaces.

We then have:

\begin{Th}[{\cite{BM2}}] \label{bmb} 

There exists a constant $C_1$
with the following property: 

\begin{enumerate}
\item[(a)]If $\Gamma$ is picked randomly from
$\calF_{n,4}^!$ for $n$ odd, then with probability $\to 1$ as $n \to
\infty$, there is an orientation $\calO$ on $\Gamma$ so that for any
splitting of $(\Gamma, \calO)$ to a $3$-regular graph $(\Gamma',
\calO')$, the surface $S^C(\Gamma', \calO')$ satisfies:
\begin{enumerate}
\item[(i)] $S^C(\Gamma', \calO')$ has genus ${\frac{n+1}{2}}$;
\item[(ii)] $\lambda_1(S^C(\Gamma', \calO')) \ge C_1$.
\end{enumerate}
\item[(b)] If $\Gamma$ is picked randomly from $\calF_{n,3}^!$ with $n
\equiv 2\, (\operatorname{mod} 4)$, then
with probability $\to 1$ as $n \to \infty$, there is an orientation 
$\calO$ on $\Gamma$ such that the surface $S^C(\Gamma', \calO')$ satisfies:
\begin{enumerate}
\item[(i)] $S^C(\Gamma', \calO')$ has genus ${\frac{n+2}{4}}$;
\item[(ii)] $\lambda_1(S^C(\Gamma', \calO')) \ge C_1$.
\end{enumerate}
\end{enumerate}
\end{Th}

\begin{Th}[{\cite{BM3}}] \label{bmc}
There exist constants $C_2, C_3, C_4,$ and $C_5$ with the following
property:
if $(\Gamma, \calO)$ is picked randomly from $\calF_{n,3}^*$, then, as
$n \to \infty$, the surface $S^C(\Gamma, \calO)$ will have the
following properties, with probability at least $C_2$:

\begin{enumerate}
\item[(a)] $\lambda_1(S^C(\Gamma, \calO)) \ge C_3$.
\item[(b)] The length of the shortest geodesic $\syst(S^C(\Gamma,
\calO))$ of $S^C(\Gamma, \calO)$ satisfies
\[\syst(S^C(\Gamma, \calO)) \ge C_4.\]
\item[(c)] The diameter $\diam(S^C(\Gamma,\calO))$ satisfies
\[\diam(S^C(\Gamma, \calO)) \le C_5 \log(\genus(S^C(\Gamma,
\calO))).\]
\end{enumerate}
\end{Th}

\section*{Acknowledgements}

The first author would like to thank the UMPA of the
\'Ecole Normale Sup\'erieure of Lyon, for its hospitality during the
time when this paper was written.

\section{Open and closed Riemann surfaces}

Let $S^O$ be an open Riemann surface, carrying a complete metric of
constant curvature $-1$ and finite area. Then there is a unique
compactification $S^C$ of $S^O$ whose conformal structure is uniquely
determined from $S^O$. In general, $S^C$ need not carry a hyperbolic
metric, but under favorable circumstances it will, and indeed the
hyperbolic metrics on $S^O$ and $S^C$ will be closely related.

\begin{Th}[{\cite{PS}}] For all $\eps$, there exists $L$ such that, if
all the cusps of $S^O$ have length $\ge L$, then there are canonically
defined cusp neighborhoods $\{\calU_i\}$ on $S^O$ and $S^C$ such that the
hyperbolic metrics $ds^2_O$ and $ds^2_C$ satisfy
\[{\frac{1}{(1+\eps)}}ds^2_O \le ds^2_C \le (1 + \eps) ds^2_O\]
outside the sets $\{ \calU_i \}$.
\end{Th}

See \cite{PS} for a precise statement. The length of a cusp is the
length of the longest closed horocycle about the cusp.

The inverse process was described in \cite{BM1}:

\begin{Th}[{\cite{BM1}}] Given $L$, there exisis a number $R$ with the
following property:
If $S$ is a compact Riemann surface, and $\{p_1, \dots, p_k\}$
points on $S$ such that
\begin{enumerate}
\item[(a)] The injectivity radius about each point is at least $R$,
and
\item[(b)] The balls $B(p_i, R)$ of radius $R$ about the points $p_i$
are pairwise disjoint, 
\end{enumerate}
then $S-\{p_1, \dots, p_k\}$ has cusps of length $\ge L$.
\end{Th}

Using this, one shows:

\begin{Th}[{\cite{PS, BM1}}] \label{comp}
For $L$ sufficiently large, there is a
constant $C(L)$ such that:
\begin{enumerate}
\item[(i)] The Cheeger  constants $h(S^O)$ and $h(S^C)$ satisfy:
\[{\frac{1}{C(L)}} h(S^O) \le h(S^C) \le C(L) h(S^O).\]
\item[(ii)] The first eigenvalues $\lambda_1(S^O)$ and
$\lambda_1(S^C)$ satisfy
\[{\frac{1}{C(L)}} \lambda_1(S^O) \le \lambda_1(S^C) \le C(L)
\lambda_1(S^O).\]
\item[(iii)] The shortest closed geodesics satisfy
\[{\frac{1}{C(L)}} \syst(S^O) \le \syst(S^C) \le C(L) \syst(S^O).\]
\end{enumerate}

Furthermore, $C(L) \to 1$ as $L \to \infty$.
\end{Th}

By combining this result with the technique of \cite{BBD} of closing
off cusps by forming handles, we prove

\begin{Th}[{\cite{BM1}}] Let $\{S_i\}$ be a collection of compact
Riemann surfaces with the following properties:
\begin{enumerate}
\item[(a)] There exists $\lambda >0$ such that $\lambda_1(S_i) >
\lambda$ for all $i$.
\item[(b)] $\syst(S_i) \to \infty$ as $i \to \infty$.
\item[(c)] For every $C >1$ there exists an $N$ such that
\[\{ x \in \R : x >N \} \subset \bigcup_i\ [1, C] (\genus(S_i)).\]
\end{enumerate}

Then, for all $\eps$, there exists $N$ such that for $g \ge N$ there
exists a surface $S_g$ of genus $g$ with 
\[\lambda_1(S_g) \ge \lambda - \eps.\]
\end{Th}

Applying this theorem to the compactifications of the modular surfaces
then gives Theorem \ref{bma}.

\section{Riemann surfaces and $3$-regular graphs}

Let $\Gamma$ be a $3$-regular graph. An {\em orientation} $\calO$ of
$\Gamma$ is an assignment to each vertex $v$ of $\Gamma$ 
of a cyclic ordering of the
edges emanating from $v$. It is clear that a $3$-regular graph on $n$ vertices
possesses $2^n$ orientations.

To the pair $(\Gamma, \calO)$ we will assign two surfaces $S^O(\Gamma,
\calO)$ and $S^C(\Gamma, \calO)$. The surface $S^O(\Gamma, \calO)$ is
obtained by gluing one copy of the ideal hyperbolic triangle $T$ shown
in Figure \ref{triangle} for each vertex of $\Gamma$, such that
the natural orientation of the geodesic segments on $T$ matches up
with the orientation about the vertex. Whenever two vertices are
joined by an edge, we glue the corresponding triangles together,
subject to the following conditions:

\begin{figure}
\includegraphics[scale=.40]{era64el-fig-1}
\caption{The marked ideal triangle $T$.}\label{triangle}
\end{figure}

\begin{enumerate}
\item[(i)] The geodesic segments on the copies of $T$ are glued
together.
\item[(ii)] The orientations on the copies of $T$ (as complex manifolds
with boundary) are preserved.
\end{enumerate}

The surfaces $S^C(\Gamma,\calO)$ are the compactifications of the
surfaces $S^O(\Gamma, \calO)$.

As discussed in \cite{TS}, the geometry and even the topology of the
surfaces $S^O(\Gamma, \calO)$ and $S^C(\Gamma, \calO)$ depend very
strongly on the orientation $\calO$. We will say that a path $\gamma$
on $(\Gamma, \calO)$ is a left-hand-turn path if, whenever it arrives
at a vertex, it turns left according to the orientation $\calO$. Then
each cusp of $S^O(\Gamma, \calO)$ is associated to a unique
left-hand-turn path. If we
denote by $\#(LHT)$ the number of these paths, then the genus of
$S^C(\Gamma, \calO)$ is clearly
\[\genus(S^C(\Gamma, \calO)) = 1 + {\frac{n - 2\#(LHT)}{4}}.\]
The length of the cusp corresponding to a given left-hand-turn path
$\gamma$
is precisely the number of edges in $\gamma$.

Many geometric properties of the surface $S^O(\Gamma, \calO)$ are
reflected in the pair $(\Gamma, \calO)$. Some of these properties
follow from general properties of covering manifolds (\cite{SGTC} and
\cite{VD}). Other properties depend more delicately on this particular
construction. When the graph has no short left-hand-turn paths, then
these properties descend to properties on $S^C(\Gamma, \calO)$ via
Theorem \ref{comp}.

\begin{Th} For some $L$ sufficiently large, there are constants $C_1,
\dots, C_6$ with the following property: Suppose that $(\Gamma,
\calO)$ has no left-hand-turn paths of length $\le L$. Then

\begin{enumerate}
\item[(i)] $ C_1 \lambda_1(\Gamma) \le \lambda_1(S^C(\Gamma,\calO)) \le
C_2 \lambda_1(\Gamma)$.
\item[(ii)] $C_3 h(\Gamma) \le S^C(\Gamma, \calO) \le C_4 h(\Gamma)$.
\item[(iii)] $\log(\syst(\Gamma)) \le \syst(S^C(\Gamma, \calO)) \le C_5
\syst(\Gamma)$.
\item[(iv)] \textup{(\cite{BM2})} $\diam(S^C(\Gamma, \calO)) \le C_6
\diam(\Gamma)$.
\end{enumerate}
\end{Th}

With such strong control over the surface $S^C(\Gamma, \calO)$, one
might be led to expect that the surfaces $S^C(\Gamma, \calO)$ are
rather rare. It is therefore rather surprising that they in fact are quite
common.

\begin{Th}[{\cite{Be, BM3}}] Given any compact Riemann surface $S$,
there are arbitrarily small deformations $S_{\eps}$ of $S$ such that
$S_{\eps} = S^C(\Gamma, \calO)$ for some pair $(\Gamma, \calO)$.
\end{Th}


\section{Models of random graphs}

Theorems \ref{bmb} and \ref{bmc} are now obtained by an analysis of
the process of picking a random graph. To carry out this analysis, we
make use of the model of random graphs considered by Bollob\'as
\cite{bo1,Bo2}. In this model, a $k$-regular graph on $n$ vertices is
constructed at random by putting $nk$ balls into a hat, $k$ balls for
each vertex. The balls are drawn out of the hat in pairs, and an edge
drawn between $v_1$ and $v_2$ each time a pair of balls corresponding
to $v_1$ and $v_2$ is drawn. An orientation on the graph may be
determined by the order in which the corresponding pairs are drawn.

We will need the following results of \cite{bo1} and \cite{Bo2}:

\begin{Th}
\begin{enumerate}
\item[(i)] \textup{(\cite{Bo2})} There is a constant $C_1$ such that, as $n \to
\infty$, the probability that $H(\Gamma) \ge C_1$ tends to $1$.
\item[(ii)] \textup{(\cite{bo1})}
Let $X_1, \dots, X_L$ denote the random variable
\[X_j= \ {\hbox{the number of closed paths of length $j$ in}}\
\Gamma.\]
\end{enumerate}

Then, for $L$ fixed and $n \to \infty$, the variables $X_1, \dots,
X_L$ tend to independent Poisson distributions.
\end{Th}

To establish Theorem \ref{bmb} (a), we seek the probability that a
randomly chosen $4$-regular graph will have an orientation with
precisely one left-hand-turn path. Using ideas of \cite{Xu1,Xu2}, it is
shown in \cite{BM2} that this will happen with probability $\to 1$ as
long as $\Gamma$ has no closed loops of length 1. The proof of
Theorem \ref{bmb} (b) is similar, using \cite{Th} in place of \cite{Xu1,Xu2}.

To establish Theorem \ref{bmc}, we estimate the probability that the
pair $(\Gamma, \calO)$ has no left-hand-turn paths of length $\le
L$. This will certainly be the case if it has no closed paths of
length $\le L$ whatsoever, from which Theorem \ref{bmc} follows. By
refining this argument, we may get substantially better estimates for
the constant $C_2$.

 



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\end{thebibliography}
\end{document}