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%\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996}
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\theoremstyle{plain}
\newtheorem*{theorem1}{Theorem}
\newtheorem*{theorem2}{Corollary}
\newtheorem*{theorem3}{Lemma} % (Lickorish)}

\theoremstyle{remark}
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\theoremstyle{definition}
\newtheorem*{definition1}{Example 1}
\newtheorem*{definition2}{Example 2}
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\begin{document}

\title[Geodesic length functions and Teichm\"{u}ller spaces]{Geodesic 
length functions\\ and Teichm\"{u}ller spaces }
\author{Feng Luo}
\address{Department of Mathematics, Rutgers University, New 
Brunswick, NJ 08903}
\email{fluo@math.rutgers.edu }
%\issueinfo{2}{1}{}{1996}
\date{April 9, 1996}
\commby{Walter Neumann}
\subjclass{Primary 32G15, 30F60}
\keywords{Hyperbolic metrics, geodesics, Teichm\"{u}ller spaces}
\begin{abstract}Given a compact orientable surface 
$\Sigma $, let
$\mathcal{S}(\Sigma )$ be the set of isotopy classes of essential simple closed
curves in $\Sigma $. We determine a complete set of relations for a function
from $\mathcal{S}(\Sigma )$ to $\mathbf{R}$ to be the geodesic length function of a hyperbolic metric
with geodesic boundary on $\Sigma $. As a consequence, the Teichm\"{u}ller
space of hyperbolic metrics with geodesic boundary on $\Sigma $
is reconstructed from an intrinsic combinatorial structure on 
$\mathcal{S}(\Sigma )$.
This also gives a complete description of the image of Thurston's 
embedding of the Teichm\"{u}ller space. 
\end{abstract}
\maketitle


\section*{\S 1. Introduction}Let $\Sigma = \Sigma _{g,r}$ be a compact oriented  surface of genus $g$ with
$r$ boundary components.
The Teichm\"{u}ller space of isotopy classes of hyperbolic metrics with geodesic
boundary  on $\Sigma $ is denoted by $T(\Sigma )$, and the set of isotopy
classes of essential simple closed unoriented curves in $\Sigma $ is denoted by
$\mathcal{S} =$ $\mathcal{S}(\Sigma )$. 
For each $m \in T(\Sigma )$ and $\alpha \in \mathcal{S}(\Sigma )$, let $l_{m}(\alpha )$ be the length
of the geodesic representing $\alpha $.

An interesting question is to characterize the geodesic length functions
among all functions defined on $\mathcal{S}(\Sigma )$.
We announce in this note a solution to this question.

The solution is expressed in terms of an intrinsic 
combinatorial structure on $\mathcal{S}(\Sigma )$. 

Before stating the theorem, let us consider three basic examples which
motivate the solution. 
We denote the isotopy class of a curve $s$ by $[s]$, and
 \{$t \in \mathbf{R} | t >a$\} by $\mathbf{R}_{ >a}$.

\begin{definition1} If the surface  is the three-holed sphere
$\Sigma _{0,3}$, then the set $\mathcal{S}(\Sigma _{0,3})$ consists
of three elements which are the isotopy classes of the three
boundary components of the surface. It is well known from the work of
Fricke-Klein \cite{FK} that any positive function from $\mathcal{S}(\Sigma )$ 
to $\mathbf{R}_{>0}$ is the
geodesic length function of a unique element in the Teichm\"{u}ller
space.

For the rest of the note, we introduce the trace function
$t_{m} (\alpha ) = $ 2cosh$l_{m}(\alpha )/2$ from $\mathcal{S}(\Sigma )$ to $\mathbf{R}_{>2}$.
We will  deal with the trace function $t_{m}$ instead of $l_{m}$. 
\end{definition1}


\begin{definition2} The surface is the one-holed torus $\Sigma _{1,1}$. 
Let $\mathcal{S}'$ be the set \{$[s] \in \mathcal{S}|$ $s$ is not homotopic to the
boundary $\partial \Sigma _{1,1}\}$. It is well known that $\mathcal{S}'$
is in one-one correspondence with the set of rational numbers
$\mathbf{Q} \cup \{\infty \}$ where the map sends the isotopy class $[s]$ to the
``slope" of $[s]$.   Two rational numbers $p/q$ and $p'/q'$ satisfying
$pq'-p'q= \pm 1$ correspond to  two isotopy classes which
contain two simple closed curves $a$ and $b$
intersecting  at one point transversely. Thus the modular
configuration comes in as a combinatorial structure on $\mathcal{S}'$.
A result of Fricke-Klein proved by L. Keen \cite{Ke} gives a solution to
the characterization problem. Namely,  a function
$f : \mathcal{S}(\Sigma _{1,1}) \to \mathbf{R}_{>2}$ 
is the trace $t_{m}$ for
some $m$ in the Teichm\"{u}ller space if and only if
\begin{equation*}f(\alpha )f(\beta )f(\gamma ) + 2 = f^{2}(\alpha ) + f^{2}(\beta ) + f^{2}(\gamma ) 
+f([\partial \Sigma _{1,1}]),\tag{1}\end{equation*}
\begin{equation*}f(\gamma ) + f(\gamma ') = f(\alpha ) + f(\beta),
\tag{1$'$}\end{equation*}
where $(\alpha , \beta , \gamma )$ and $(\alpha , \beta , \gamma ')$ are
two ideal triangles in the modular configuration. 
\end{definition2}

%\vskip 4in %\centerline{\epsfxsize=4in\epsfbox{1.ps}}
\begin{figure}[t]
\includegraphics[scale=.6]{era8e-fig-1}
\caption{}
\end{figure}
\begin{definition3} The surface is the four-holed sphere $\Sigma _{0,4}$.
Let $\mathcal{S}'$ be the set \{$[s] \in \mathcal{S}|$ $s$ is 
not homotopic to the
boundary $\partial \Sigma _{0,4}$\}. It is well known  \cite{De}, \cite{Th}
that $\mathcal{S}'$ is in one-one correspondence with  the
set of rational numbers $\mathbf{Q} \cup \{\infty \}$ so that two rational
numbers $p/q$ and $p'/q'$ satisfy $pq'-p'q= \pm 1$ if and only if
the isotopy classes $p/q$ and $p'/q'$ are distinct and
 contain  two simple closed curves
$a$ and $b$ which intersect at two points.
Thus again the modular configuration comes in as a 
combinatorial structure on $\mathcal{S}'$.
It is shown in \cite{Lu1} that  a function
$f : \mathcal{S}(\Sigma _{0,4}) \to \mathbf{R}_{>2}$ is the trace $t_{m}$ for 
some $m$ in the Teichm\"{u}ller space if and only if
\begin{equation*}\begin{split}
& f(\alpha )f(\beta )f(\gamma ) + 4\\ 
&\qquad= f^{2}(\alpha ) + f^{2}(\beta ) + f^{2}(\gamma ) +
f(\alpha )( f(b_{i})f(b_{j}) + f(b_{k}) f(b_{l}))\\ 
 &\qquad\quad+ f(\beta )( f(b_{i})f(b_{k}) + f(b_{j}) f(b_{l}))
+ f(\gamma )( f(b_{i})f(b_{l}) + f(b_{j})f(b_{k})) \\
&\qquad\quad+ \sum _{s=1}^{4} f^{2}(b_{s}) + f(b_{1})
f(b_{2})f(b_{3})f(b_{4}), \quad i \neq j \neq k \neq i,
\end{split}\tag{2}
\end{equation*}
\begin{equation*}f(\gamma ) + f(\gamma ') = f(\alpha ) f(\beta ) - f(b_{i}) f(b_{l})
- f(b_{j}) f(b_{k})\tag{2$'$}
\end{equation*}
where $(\alpha , \beta , \gamma )$ and $(\alpha , \beta , \gamma ')$ are
two ideal triangles in the modular configuration,
and $b_{s}$'s ($s=1,2,3,4$)
are the isotopy classes of the four boundary components of $\partial \Sigma _{0,4
}$ so that each of the triples \{$\alpha $, $b_{i}$, $b_{j}$\},
\{$\beta $, $b_{i}$, $b_{k}$\} and \{$ \gamma , b_{i}, b_{l}$\}
bounds a 3-holed sphere in $\Sigma $.
\end{definition3}


Our main result states that relations (1), ($1'$), (2) and ($2'$)
are the set of all
relations for a function from $\mathcal{S}(\Sigma )$ to $\mathbf{R}_{>2}$
to be the trace function $t_{m}$ for an element $m$ in the Teichm\"{u}ller space.

To be more precise, we introduce a combinatorial structure
(corresponding to the modular configuration) on $\mathcal{S}$ as follows.
Given two isotopy classes $\alpha $ and $\beta $,
let I($\alpha , \beta $) be the geometric
intersection number between $\alpha $ and $\beta $ in $\mathcal{S}(\Sigma )$, i.e.,
I($\alpha , \beta $) = Min\{$| a \cap b| \mid $ $a 
\in \alpha $ and $b \in \beta $\},
where $|a \cap b|$  is the number of points in $a \cap b$.
If two simple closed curves $a$ and
$b$ intersect at one point transversely (resp. $\alpha $, $\beta \in \mathcal{S}(\Sigma )$
with I($\alpha , \beta $) = 1), 
we denote it by $a \perp b$ (resp. $\alpha \perp \beta $); if two simple
closed curves $a$ and $b$ intersect at two points of different signs
transversely  and I($[a], [b]$) = 2, 
we denote it by $a \perp _{0} b$. In this case,  we  denote the relation
between their isotopy classes by $[a] \perp _{0} [b]$.
Suppose $x$ and $y$ are two arcs in $\Sigma $ so that $x$ intersects $y$
transversely at one point. Then \em the resolution of $x \cap y$ from
$x$ to $y$ \rm is defined as follows. Take any orientation on $x$ and
use the orientation on $\Sigma $ to determine an orientation on $y$. Now
resolve the intersection point $x \cap y$ according to the orientations
(see Figure 2(a)).
If $a \perp b$  or $a \perp _{0} b$, we define $ab$ to be the
curve obtained  by resolving intersection points in $a \cap b$
from $a$ to $b$.  
If $\alpha \perp \beta $ or $\alpha \perp _{0} \beta $, we define
$\alpha \beta $ to be $[ab]$ where $a \in \alpha $ and $b \in \beta $ with
$|a \cap b| =$ I$(\alpha , \beta $). Note that in the examples 2 and 3,
$(\gamma , \gamma ')$ is $(\alpha \beta , \beta \alpha )$. 

\begin{theorem1}  For a surface $\Sigma _{g, r}$ of negative Euler
number, a function $l$ from $\mathcal{S}(\Sigma _{g,r})$ to $\mathbf{R}_{t >0}$ 
is the geodesic length function of a hyperbolic
metric with geodesic boundary  on
$\Sigma _{g,r}$ if and only
if 

(a) for any  two simple closed curves $a_{1}$, $a_{2}$  with $a_{1} \perp a_{2}$, let
$a_{3} = a_{1} a_{2}$ and $b$ be the boundary of a regular neighborhood of
$a_{1} \cup a_{2}$, then
\begin{gather} t_{1} t_{2} t_{3}  + 2 
= t_{1}^{2} + t_{2}^{2} + t_{3}^{2}  + t([b]), \notag\\
t_{3} + t_{3}' = t_{1} t_{2},\notag
\end{gather}
where $t_{i} = 2\cosh(l([a_{i}])/2)\;(i=1,2,3)$,  $t_{3}' = 
2\cosh(l([a_{2}a_{1}])/2)$,
and $t([b]) =  2\cosh (l([b])/2)$,
and

(b) for any  two simple closed curves  $a_{12}$ and $a_{23}$ with
$a_{12} \perp _{0} a_{23}$, let $a_{31} = a_{12} a_{23}$ and
$b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$  be the four boundary  components of a regular neighborhood
of $a_{12} \cup a_{23}$ so that  $a_{ij}$, $b_{i}$ and $b_{j}$ bound a
subsurface $\Sigma _{0,3}\; (i,j \leq 3)$, then
\vbox {\begin{equation*}\begin{split}
t_{12}t_{23}t_{31} + 4 & = t_{12}^{2} + t_{23}^{2} + t_{31}^{2} +
 t_{12}( t_{1} t_{2} + t_{3} t_{4}) + t_{23}( t_{2} t_{3} +
 t_{1} t_{4})
\\ & \qquad + t_{31}( t_{3} t_{1} + t_{2} t_{4})
 + t_{1}^{2} + t_{2}^{2} + t_{3}^{2} + t_{4}^{2} +  t_{1}t_{2}t_{3}t_{4},
\end{split}\end{equation*}
\begin{equation*} t_{31} + t_{31}' 
= t_{12}t_{23} - t_{1}t_{3} - t_{2} t_{4},
\end{equation*}
where $t_{i} = 2\cosh(l([b_{i}])/2),\; t_{ij} = 2\cosh(l([a_{ij}])/2)$, 
and $t_{31}' = 2\cosh(l([a_{23}a_{12}])/2)$.}
\end{theorem1}

%\vskip 5in  %\centerline{\epsfxsize=5in\epsfbox{2.ps}}
\begin{figure}[t]
\includegraphics[scale=.55]{era8e-fig-2}
\caption{}
\end{figure}


Figure 2 (surfaces have the right-hand orientations in the
front face) shows the location of these curves. Note that the curves
$a_{i}$ and $a_{ij}$ are symmetric in the sense that 
$[a_{i}][ a_{j}] = [a_{k}]$ and $[a_{ij}][ a_{jk}] = [a_{ki}]$  for $k \neq i,j$ and $(i,j) = (1,2), (2,3), (3,1)$.

Relations (1), ($1'$), (2), and ($2'$)
come from trace identities for SL(2,$\mathbf{R}$) matrices.

Thurston's  compactification of the Teichm\"{u}ller space  $T(\Sigma )$ (see
\cite{ Bo}, \cite{FLP}, \cite{Th}) uses  the embedding $\pi : T(\Sigma ) \to \mathbf{R}^{\mathcal{S}(\Sigma )}$ sending $m$
to $l_{m}$. The theorem gives a complete description of the image of the
embedding.

The proof of the theorem also shows the following result.
Given a  
subset $F$ of $\mathcal{S}(\Sigma )$, 
let $\pi _{F} : T(\Sigma ) \to \mathbf{R}^{F}$ be the map
$\pi _{F} (m) = l_{m} |_{F}$.

\begin{theorem2} (a) 
For a surface $\Sigma _{g,r}$ of negative Euler number and $r >0$, there
exists a finite subset $F$ in $\mathcal{S}(\Sigma _{g,r})$ consisting of
$6g + 3r -6$ elements so that the map $\pi _{F} : T(\Sigma _{g,r})
\to \mathbf{R}^{F}$ is a real analytic embedding onto an open subset which is
defined by a finite set of (explicit) real analytic inequalities in the
coordinates of $\pi _{F}$.

(b) For a surface $\Sigma _{g, 0}$ of negative Euler number, there exists
a finite subset $F$ of $\mathcal{S}(\Sigma _{g,0})$ consisting of $6g-5$
elements so that $\pi _{F} : T(\Sigma _{g,0}) \to \mathbf{R}^{F}$ 
is an embedding whose image in $\mathbf{R}^{F}$ is defined
by one real analytic equation and finitely
 many (explicit) real analytic inequalities in the coordinates of $\pi _{F}$.
\end{theorem2}
It is  shown by S. Wolpert \cite{Wo} that the number  $6g-5$ in part (b) of the
 corollary is minimal. The corollary without the statements about the image of
the embedding was obtained previously by Okumura \cite{Ok}, Schmutz \cite{Sc},
and Sorvali \cite{So}. Okumura \cite{Ok1} has recently proven  the
corollary using a different method. 

Some examples of the collection $F$ and the
images of the Teichm\"{u}ller spaces are  as follows.
For $\Sigma _{2,0}$, 
take $F =\{[a_{1}], [a_{2}], [a_{3}],
[a_{4}], [a_{5}], [a_{6}], [a_{7}]\}$ as in Figure 3. 
Then the  map $\pi _{F}$ is an embedding with image 
$\pi _{F}(T(\Sigma _{2,0}^{0}))$  $=\{(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7})
\in \mathbf{R}_{>2}^{6}$ $|$ $t_{8} > 2$, $t_{9} > 2$, $t_{8} = t_{6}t_{7}t_{9} -t_{6}^{2}
-t_{7}^{2} - t_{9}^{2} + 2$, where $t_{8} = t_{1}t_{2}t_{3} -t_{1}^{2} - t_{2}^{2} -t_{3}^{2} +2$,
and $(2 + t_{2}^{2} + t_{8}) t_{9}^{2} + 2t_{2}(t_{4} + t_{5}) t_{9}$ $+ 2t_{2}^{2} + t_{4}^{2} + t_{5}^{2} + t_{8}^{2} + t_{2}^{2}t_{8}$ $-t_{4}t_{5}t_{8} -4$ $= 0$\}. 

%\vskip 5in %\centerline{\epsfxsize=5in\epsfbox{3.ps}}
\begin{figure}[t]
\includegraphics[scale=.45]{era8e-fig-3}
\caption{}
\end{figure}

\begin{remark1} The theorem and the corollary hold 
for hyperbolic metrics with cusp ends (see \cite{Lu1}). 
\end{remark1}


\section*{Acknowledgment}I would like to thank F. Bonahon  
for calling my attention
to several references. This work is supported in part by the NSF.

\section*{\S 2. Sketch of the proof of the theorem}

There are several basic steps involved in the proof of the theorem.
 The proof is  currently quite long. Below,
we shall describe briefly the idea of the proof for a
compact surface with  non-empty boundary. 

Step 1. We prove the  result stated in Example 2 for $\Sigma _{0,4}$. 
This is essentially
a careful application of the Maskit combination theorem \cite{Mas}
together with the
trace relations for SL(2,$\mathbf{R}$) matrices.

Step 2. We prove a gluing lemma. The classical gluing lemma
for surfaces is as follows. Take two  hyperbolic surfaces $X$ and
$Y$ with geodesic boundary so that they have the same lengths at
two boundary components $b_{X}$ and $b_{Y}$. Then one glues $X$ to $Y$ along
$b_{X}$ and $b_{Y}$ by an isometry. However, the isometry is not unique due
to the rotational symmetry of $S^{1}$. One obtains the so-called 
Fenchel-Nielsen twisting parameter for the gluing. We propose another
procedure of gluing which will eliminate the parameter. Thus the
result of the gluing produces a unique hyperbolic metric up to
isotopy. The basic idea is as follows. Given a compact surface
$\Sigma $, we decompose it as a union of two compact connected incompressible
(meaning $\pi _{1}$-injective) subsurfaces $X$ and 
$Y$ so that $ X \cap Y$ is homeomorphic to
$\Sigma _{0,3}$ (see Figure 4). 
Let the three boundary components of $X \cap Y$
be $a_{1}$, $a_{2}$ and $a_{3}$.
Then the gluing lemma states that for
each hyperbolic metric $m_{X}$ and $m_{Y}$ on $X$ and $Y$ respectively
so that $a_{i}$ are geodesics in both metrics with 
$l_{m_{X}}( a_{i}) = l_{m_{Y}}(a_{i}),\;
(i=1,2,3)$ there is a hyperbolic
metric $m$ in $\Sigma $ unique up to isotopy so that the restriction
of $m$ to $X$ is isotopic to $m_{X}$ and the restriction of $m$ to $Y$ is
isotopic to $m_{Y}$.  
The proof of the lemma is evident from the definition.
For simplicity, we shall call this the gluing along 3-holed sphere lemma. 

Step 3.  We use the gluing along 3-holed sphere
lemma to understand the hyperbolic metrics
on $\Sigma _{1,2}$ by decomposing  $\Sigma _{1,2}$ as a union $X \cup Y$,
 where
$X \cong \Sigma _{1,1}$ and $Y \cong \Sigma _{0,4} -(b_{1} \cup b_{2})$,
where $b_{1}$ and $b_{2}$ are the two boundary components.
See Figure 4(c). A slight generalization of the version of
the gluing lemma stated above is needed to take care of the non-compact $Y$.

%\vskip 5in %\centerline{\epsfxsize=5in\epsfbox{4.ps}}
\begin{figure}[t]
\includegraphics[scale=.55]{era8e-fig-4}
\caption{}
\end{figure}

Step 4. We set up the induction procedure as follows. Define the
norm of a surface $\Sigma _{g,r}$ to be $3g+r$. Given
a surface $\Sigma _{g,r}$, if  $r \geq 2$,
then $\Sigma _{g,r} = X \cup Y$, where $X =\Sigma _{g,r-1}$
and $Y =\Sigma _{0,4}$, and if $r=1$,
$\Sigma _{g,1} = X \cup Y$, where $X = \Sigma _{g-1, 2}$ and
$Y = \Sigma _{1,2}$  with $X \cap Y \cong \Sigma _{0,3}$ (see Figure 4).
Thus given a function $f$ from $\mathcal{S}(\Sigma _{g,r})$  to $\mathbf{R}_{>2}$
satisfying conditions (1), ($1'$), (2), and ($2'$), 
we consider the restrictions of
$f$ to $\mathcal{S}(X)$ and $\mathcal{S}(Y)$ (both are viewed as subsets
of $\mathcal{S}(\Sigma )$ under the inclusion maps). By the
induction hypothesis, there exist  hyperbolic metrics $m_{X}$ and $m_{Y}$
on $X$ and $Y$, respectively, 
so that $f |_{\mathcal{S}(X)} = t_{m_{X}}$ and
$f|_{\mathcal{S}(Y)} = t_{m_{Y}}$. Now by the gluing lemma, there
exits a metric $m$ on $\Sigma $ whose restriction to the two
subsurfaces $X$ and $Y$ are isotopic to $m_{X}$ and $m_{Y}$. Thus,
we have constructed a hyperbolic metric $m$ on $\Sigma $ so that $f$ is
equal to $t_{m}$ on the subset $\mathcal{S}(X)
\cup \mathcal{S}(Y)$  of $\mathcal{S}(\Sigma )$.

Step 5.  This is the key step in the proof. The goal is to show that the above
condition  $f|_{\mathcal{S}(X) \cup \mathcal{S}(Y)} = t_{m}|_{\mathcal{S}(X) \cup \mathcal{S}(Y)}$
implies $f = t_{m}$. 

Our observation is that the equations ($1'$) and ($2'$) give rise to
an iteration process.

This shows that the value of $f$ at $\beta \alpha $ is
determined by the values of $f$ at  $\alpha $, $\beta $, $\alpha \beta $
in case $\alpha \perp \beta $, and the values of $f$ at $\alpha $, $\beta $,
$\alpha \beta $ 
and $ b_{s}$'s in case $\alpha \perp _{0} \beta $.
For instance, to determine $f$ on $\mathcal{S}'(\Sigma _{1,1})$,
it suffices to know the value of $f$ at three vertices of an ideal
triangulation in the modular configuration since the iteration
equation ($1'$) will take care of the values of $f$ on $\mathcal{S}'$.

We shall illustrate the proof of this major step by considering the 
special
case of $\Sigma =\Sigma _{0,5}$. Take two disjoint, essential,
non-boundary parallel, non-homotopic simple
closed curves $a_{1}$ and $a_{2}$ in $\Sigma $. They give rise to a
gluing along a 3-holed sphere decomposition of $\Sigma $, where
we take $X_{i}$ to be the 4-holed sphere bounded by $a_{i}$ in $\Sigma $.

Now the given condition on $f$ and $t_{m}$ states that $f([a])
= t_{m}([a])$ for all simple closed curves $a$ which are disjoint from
either $a_{1}$ or $a_{2}$.  A lemma of Lickorish below shows that
if $f([a]) = t_{m}([a])$ holds for all simple closed curves $a$
which intersect each $a_{i}$ in at most two points, then $f = t_{m}$.

\begin{theorem3}[Lickorish] Suppose $s$ is an essential simple closed curve
which intersects one of the two curves $a_{i}$ in  at least three points.
Let the norm of a curve $c$ be $|c \cap a_{1}| + |c \cap a_{2}|$.
Then there exist two simple closed curves $p$ and $q$ with $p \perp _{0} q$
so that $s = pq$
and the norms of the curves $p$, $q$, $qp$, and each of the
four boundary components of a regular neighborhood $N(p \cup q)$
of $p \cup q$ are less than the norm of $s$.
\end{theorem3}
We sketch the proof of this lemma as follows. 
Suppose for definiteness that $|s \cap a_{1}|> 2$. Then since 
$s$ is a separating simple closed curve, the intersection points of
$s \cap a_{1}$ have alternating intersection signs in $a_{1}$. Pick up
three adjacent intersection points $x$, $y$ and $z$ in $a_{1}$ and fix
an orientation on $s$. Without loss of generality, we may assume that 
the arc from $x$ to $y$ (in curve $s$) in the orientation does not contain the point $z$.
Then choose simple closed curves $p$ and $q$
as indicated in Figure 5. One verifies that the $p$, $q$  and
$\partial N(p \cup q)$
satisfy the
condition in the lemma (see \cite{Lu2} for more details on curves in
surfaces).

%\vskip 5in %\centerline{\epsfxsize=5in\epsfbox{5.ps}}
\begin{figure}[t]
\includegraphics[scale=.41]{era8e-fig-5}
\caption{}
\end{figure}


Thus, to finish the proof of the theorem, by the lemma above and the
iterate equation ($2'$), it suffices to show
that $f([a]) = t_{m}([a])$ where $a$ intersects each $a_{i}$ at two points.
There are only finitely many such $a$ up to homeomorphisms of the
surface leaving each $a_{i}$ invariant. 
We verify the last condition  $f([a]) = t_{m}([a])$  for $a \perp _{0} a_{i}$
($i=1,2$) through iterated uses of the relations (2) and ($2'$).

This finishes the sketch of the proof.

\bibliographystyle{amsalpha}
\begin{thebibliography}{[FLP]}

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\bibitem[Li]{Li}
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\end{thebibliography}

\end{document}