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To display a typeset version of a journal * % * article easily, we suggest that you either view the HTML version or * % * retrieve the article in DVI, PostScript, or PDF format. * % ************************************************************************** % Author Package %% Translation via Omnimark script a2l, June 27, 1996 (all in one day!) %\controldates{16-JUL-1996,16-JUL-1996,16-JUL-1996,23-JUL-1996} \documentclass{era-l} \usepackage{graphicx} %\issueinfo{2}{1}{}{1996} %% Declarations: \theoremstyle{plain} \newtheorem*{theorem1}{Theorem} \newtheorem*{theorem2}{Corollary} \newtheorem*{theorem3}{Lemma} % (Lickorish)} \theoremstyle{remark} \newtheorem*{remark1}{Remark} \theoremstyle{definition} \newtheorem*{definition1}{Example 1} \newtheorem*{definition2}{Example 2} \newtheorem*{definition3}{Example 3} %% User definitions: \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]} \bibitem[Bo]{Bo} F. Bonahon, {\em The geometry of Teichm\"{u}ller space via geodesic currents}, % vol.~92, Invent. 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