ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 1, August 1996

GEODESIC LENGTH FUNCTIONS AND TEICHMÜLLER SPACES

FENG LUO

(Communicated by Walter Neumann)

Abstract. Given a compact orientable surface Σ, let S(Σ) be the set of isotopy classes of essential simple closed curves in Σ. We determine a complete set of relations for a function from S(Σ) to R to be the geodesic length function of a hyperbolic metric with geodesic boundary on Σ. As a consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary on Σ is reconstructed from an intrinsic combinatorial structure on S(Σ). This also gives a complete description of the image of Thurston’s embedding of the Teichmüller space.

§1. Introduction

Let Σ = Σ_{g,r} be a compact oriented surface of genus g with r boundary components. The Teichmüller space of isotopy classes of hyperbolic metrics with geodesic boundary on Σ is denoted by T (Σ), and the set of isotopy classes of essential simple closed unoriented curves in Σ is denoted by S = S(Σ). For each m ∈ T (Σ) and α ∈ S(Σ), let l_{m}(α) be the length of the geodesic representing α.

An interesting question is to characterize the geodesic length functions among all functions defined on S(Σ). We announce in this note a solution to this question.

The solution is expressed in terms of an intrinsic combinatorial structure on S(Σ).

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 ∈ R|t > a} by R_{>a}.

Example 1. If the surface is the three-holed sphere Σ_{0,3}, then the set S(Σ_{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 [FK] that any positive function from S(Σ) to R_{>0} is the geodesic length function of a unique element in the Teichmüller space.

For the rest of the note, we introduce the trace function t_{m}(α) = 2coshl_{m}(α)/2 from S(Σ) to R_{>2}. We will deal with the trace function t_{m} instead of l_{m}.

Example 2. The surface is the one-holed torus Σ_{1,1}. Let S′ be the set {[s] ∈ S| s is not homotopic to the boundary ∂Σ_{1,1}}. It is well known that S′ is in oneone correspondence with the set of rational numbers Q ∪ {∞} where the map

\begin{matrix}c \\ ◯ \end{matrix}1996 American Mathematical Society

Received by the editors April 9, 1996.

1991 Mathematics Subject Classification. Primary 32G15, 30F60.

Key words and phrases. Hyperbolic metrics, geodesics, Teichmüller spaces.

Figure 1

sends the isotopy class [s] to the “slope” of [s]. Two rational numbers p/q and p′/q′ satisfying pq′−p′q = ±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 S′. A result of Fricke-Klein proved by L. Keen [Ke] gives a solution to the characterization problem. Namely, a function f : S(Σ_{1,1}) → R_{>2} is the trace t_{m} for some m in the Teichmüller space if and only if

f(α)f(β)f(γ) + 2 = f^{2}(α) + f^{2}(β) + f^{2}(γ) + f([∂Σ_{1,1}]), (1)

f(γ) + f(γ′) = f(α) + f(β), (1′)

where (α, β, γ) and (α, β, γ′) are two ideal triangles in the modular configuration.

Example 3. The surface is the four-holed sphere Σ_{0,4}. Let S′ be the set {[s] ∈ S| s is not homotopic to the boundary ∂Σ_{0,4}}. It is well known [De], [Th] that S′ is in one-one correspondence with the set of rational numbers Q ∪ {∞} so that two rational numbers p/q and p′/q′ satisfy pq′ − p′q = ±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 S′. It is shown in [Lu1] that a function f : S(Σ_{0,4}) → R_{>2} is the trace t_{m} for some m in the Teichmüller space if and only if

\begin{matrix}f(α)f(β)f(γ) + 4 \\ = f^{2}(α) + f^{2}(β) + f^{2}(γ) + f(α)(f(b_{i})f(b_{j}) + f(b_{k})f(b_{l})) \\ + f(β)(f(b_{i})f(b_{k}) + f(b_{j})f(b_{l})) + f(γ)(f(b_{i})f(b_{l}) + f(b_{j})f(b_{k})) \\ + ∑_{s=1}^{4} f^{2}(b_{s}) + f(b_{1})f(b_{2})f(b_{3})f(b_{4}), i ≠ j ≠ k ≠ i, \end{matrix} (2)

f(γ) + f(γ′) = f(α)f(β) − f(b_{i})f(b_{l}) − f(b_{j})f(b_{k}) (2′)

where (α, β, γ) and (α, β, γ′) 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 ∂Σ_{0,4} so that each of the triples {α, b_{i}, b_{j}}, {β, b_{i}, b_{k}} and {γ, b_{i}, b_{l}} bounds a 3-holed sphere in Σ.

Our main result states that relations (1), (1′), (2) and (2′) are the set of all relations for a function from S(Σ) to R_{>2} to be the trace function t_{m} for an element m in the Teichmüller space.

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

Theorem. For a surface Σ_{g,r} of negative Euler number, a function l from S(Σ_{g,r}) to R_{t>0} is the geodesic length function of a hyperbolic metric with geodesic boundary on Σ_{g,r} if and only if

(a) for any two simple closed curves a_{1}, a_{2} with a_{1} ⊥ a_{2}, let a_{3} = a_{1}a_{2} and b be the boundary of a regular neighborhood of a_{1} ∪ a_{2}, then

t_{1}t_{2}t_{3} + 2 = t_{1}^{2} + t_{2}^{2} + t_{3}^{2} + t([b]),
t_{3} + t′_{3} = t_{1}t_{2},

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

Figure 2

(b) for any two simple closed curves a_{12} and a_{23} with a_{12} ⊥_{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} ∪ a_{23} so that a_{ij}, b_{i} and b_{j} bound a subsurface Σ_{0,3} (i, j ≤ 3), then

\begin{aligned}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}) \\ & + 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{aligned}

t_{31} + t′_{31} = t_{12}t_{23} − t_{1}t_{3} − t_{2}t_{4},

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).

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 ≠ i, j and (i, j) = (1, 2), (2, 3), (3, 1).

Relations (1), (1′), (2), and (2′) come from trace identities for SL(2,R) matrices. Thurston’s compactification of the Teichmüller space T (Σ) (see [Bo], [FLP], [Th])

uses the embedding π : T (Σ) → R^{S(Σ)} 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 S(Σ), let π_{F} : T (Σ) → R^{F} be the map π_{F} (m) = l_{m}|_{F}.

Corollary. (a) For a surface Σ_{g,r} of negative Euler number and r > 0, there exists a finite subset F in S(Σ_{g,r}) consisting of 6g + 3r − 6 elements so that the map π_{F} : T (Σ_{g,r}) → 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 π_{F}.

(b) For a surface Σ_{g,0} of negative Euler number, there exists a finite subset F of S(Σ_{g,0}) consisting of 6g − 5 elements so that π_{F} : T (Σ_{g,0}) → R^{F} is an embedding whose image in R^{F} is defined by one real analytic equation and finitely many (explicit) real analytic inequalities in the coordinates of π_{F}.

It is shown by S. Wolpert [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 [Ok], Schmutz [Sc], and Sorvali [So]. Okumura [Ok1] has recently proven the corollary using a different method.

Some examples of the collection F and the images of the Teichmüller spaces are as follows. For Σ_{2,0}, take F = {[a_{1}], [a_{2}], [a_{3}], [a_{4}], [a_{5}], [a_{6}], [a_{7}]} as in Figure 3. Then

Figure 3

the map π_{F} is an embedding with image π_{F} (T (Σ_{2,0}^{0})) = {(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7}) ∈ 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}.

Remark. The theorem and the corollary hold for hyperbolic metrics with cusp ends (see [Lu1]).

Acknowledgment

I would like to thank F. Bonahon for calling my attention to several references. This work is supported in part by the NSF.

§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 Σ_{0,4}. This is essentially a careful application of the Maskit combination theorem [Mas] together with the trace relations for SL(2,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 Σ, we decompose it as a union of two compact connected incompressible

Figure 4

(meaning π_{1}-injective) subsurfaces X and Y so that X ∩ Y is homeomorphic to Σ_{0,3} (see Figure 4). Let the three boundary components of X ∩ 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 Σ 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 Σ_{1,2} by decomposing Σ_{1,2} as a union X ∪ Y, where X ≅ Σ_{1,1} and Y ≅ Σ_{0,4} − (b_{1} ∪ 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.

Step 4. We set up the induction procedure as follows. Define the norm of a surface Σ_{g,r} to be 3g + r. Given a surface Σ_{g,r}, if r ≥ 2, then Σ_{g,r} = X ∪ Y, where X = Σ_{g,r−1} and Y = Σ_{0,4}, and if r = 1, Σ_{g,1} = X ∪ Y, where X = Σ_{g−1,2} and Y = Σ_{1,2} with X ∩ Y ≅ Σ_{0,3} (see Figure 4). Thus given a function f from S(Σ_{g,r}) to R_{>2} satisfying conditions (1), (1′), (2), and (2′), we consider the restrictions of f to S(X) and S(Y) (both are viewed as subsets of S(Σ) 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|_{S(X)} = t_{m_{X}} and f|_{S(Y)} = t_{m_{Y}}. Now by the gluing lemma, there exits a metric m on Σ 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 Σ so that f is equal to t_{m} on the subset S(X) ∪ S(Y) of S(Σ).

Step 5. This is the key step in the proof. The goal is to show that the above condition f|_{S(X)∪S(Y)} = t_{m}|_{S(X)∪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 βα is determined by the values of f at α, β, αβ in case α ⊥ β, and the values of f at α, β, αβ and b_{s}’s in case α ⊥_{0} β. For instance, to determine f on S′(Σ_{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 S′.

We shall illustrate the proof of this major step by considering the special case of Σ = Σ_{0,5}. Take two disjoint, essential, non-boundary parallel, non-homotopic

Figure 5

simple closed curves a_{1} and a_{2} in Σ. They give rise to a gluing along a 3-holed sphere decomposition of Σ, where we take X_{i} to be the 4-holed sphere bounded by a_{i} in Σ.

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}.

Lemma (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 ∩ a_{1}| + |c ∩ a_{2}|. Then there exist two simple closed curves p and q with p ⊥_{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 ∪ q) of p ∪ q are less than the norm of s.

We sketch the proof of this lemma as follows. Suppose for definiteness that |s ∩ a_{1}| > 2. Then since s is a separating simple closed curve, the intersection points of s ∩ 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 ∂N(p ∪ q) satisfy the condition in the lemma (see [Lu2] for more details on curves in surfaces).

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 ⊥_{0} a_{i} (i = 1, 2) through iterated uses of the relations (2) and (2′).

This finishes the sketch of the proof.

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Department of Mathematics, Rutgers University, New Brunswick, NJ 08903

   E-mail address: fluo@math.rutgers.edu