Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 75–88

LENGTH FUNCTIONS AND PARAMETERIZATIONS

OF TEICHMÜLLER SPACE FOR SURFACES WITH CUSPS

Ursula Hamenstädt

Mathematisches Institut der Universität Bonn Beringstrasse 1, D-53115 Bonn, Germany; ursula@math.uni-bonn.de

Abstract. For g ≥ 0 and m ≥ 1 such that 2g−2+m ≥ 1 let 𝒯_{g,m} be the Teichmüller space of hyperbolic metrics on a surface of genus g with m punctures, and let ∂𝒯_{g,m} be its Thurston boundary. Using geodesic length functions, we construct a homeomorphism of 𝒯_{g,m} ∪ ∂𝒯_{g,m} onto a convex finite-sided polyhedron in R P^{6g−6+2m}.

1. Introduction

A Riemann surface of finite type is a closed Riemann surface from which a finite number m ≥ 0 of points, the so-called punctures, have been deleted. Such a surface S_{0} is topologically determined by its genus g ≥ 0 and the number m of its punctures. In the sequel we only consider surfaces S_{0} of negative Euler characteristic with at least one puncture which are different from the thrice-punctured sphere. Then S_{0} carries a nontrivial family of complete hyperbolic metrics of finite volume.

The Teichmüller space 𝒯_{g,m} of marked hyperbolic metrics on S_{0} is the set of all pairs (f, h) where h is a hyperbolic metric on a surface S and f is the homotopy class of a homeomorphism F: S_{0} → S of S_{0} onto S. With respect to a natural topology, the space 𝒯_{g,m} is homeomorphic to an open cell of dimension 6g − 6 + 2m.

A geodesic length function on 𝒯_{g,m} is defined by the choice of a closed curve γ on the base surface S_{0} which is not nullhomotopic and not puncture parallel, i.e. which cannot be homotoped into one of the punctures. For every hyperbolic metric h on a surface S which is marked by the homotopy class of a homeomorphism F: S_{0} → S the curve F (γ) is then freely homotopic to a unique closed geodesic with respect to the metric h. The length l_{γ}(S) of this geodesic depends on h and the marking and defines a smooth (in fact real analytic) function l_{γ} on 𝒯_{g,m} which we call the length function of γ.

It is well known [FLP] that Teichmüller space can be parameterized by finitely many of these length functions. A natural problem is then to find a collection γ_{1}, …, γ_{k} of closed geodesics on our base surface S_{0} of minimal cardinality whose length functions define an embedding of 𝒯_{g,m} into R^{k}. For surfaces with cusps, i.e. if m ≥ 1, this problem was solved by Seppälä and Sorvali [SS2] who showed that 𝒯_{g,m} can be parameterized by 6g − 6 + 2m length functions of simple closed geodesics. For closed surfaces it is known that no embedding of 𝒯_{g,0} into R^{6g−6} by 6g − 6 geodesic length functions exists. In this case the optimal answer was given by Schmutz Schaller [S1] who found an embedding of 𝒯_{g,0} by 6g−5 geodesic length functions. In Section 4 of this note we construct a new and simpler such embedding of 𝒯_{g,0} into R^{6g−5} .

2000 Mathematics Subject Classification: Primary 30F60.

Partially supported by the Sonderforschungsbereich 256.

The Thurston boundary ∂𝒯_{g,m} of 𝒯_{g,m} consists of projective classes of measured geodesic laminations and is homeomorphic to a sphere of dimension 6g − 7 + 2m. It defines naturally a compactification of 𝒯_{g,m} in such a way that 𝒯_{g,m} ∪ ∂𝒯_{g,m} is homeomorphic to a closed ball.

To construct parameterizations of 𝒯_{g,m} which extend continuously to a homeomorphism of the Thurston boundary we have to projectivize our k -tuple of length functions. Namely, if the curves γ_{1}, …, γ_{k} fill up, i.e. if every closed geodesic on our surface intersects at least one of the curves γ_{i} transversely, then the map which assigns to a surface S ∈ 𝒯_{g,m} the projectivized k -tuple

[l_{γ_{1}}(S), …, l_{γ_{k}}(S)] ∈ R P^{k−1}

of length functions extends continuously to the Thurston boundary by mapping a measured lamination η to its projectivized k -tuple [i(η, γ_{1}), …, i(η, γ_{k})] ∈ R P^{k−1} of intersection numbers. We call a map of 𝒯_{g,m} ∪ ∂𝒯_{g,m} into R P^{k−1} geometric if it is defined in this way by geodesic length functions.

Call a subset P of the real projective space R P^{k−1} a finite-sided convex polyhedron if it is the projection of an intersection of finitely many closed halfspaces in R^{k}. We show:

Theorem. For every g ≥ 0 and m ≥ 1 there is a geometric homeomorphism of 𝒯_{g,m} ∪ ∂𝒯_{g,m} onto a finite-sided convex polyhedron in R P^{6g−6+2m}.

The case of closed surfaces seems to be much more difficult. A geometric embedding of 𝒯_{g,0} into a real projective space of minimal dimension is only known for g = 2 [S2]. In any case it can be easily computed that the projections into R P^{6g−6} of the known embeddings of 𝒯_{g,0} into R^{6g−5} do not extend to injective maps on the Thurston boundary. On the other hand, there are 6g − 5 length functions of simple closed geodesics which define a homeomorphism of the Thurston boundary ∂𝒯_{g,0} onto the boundary of a finite-sided convex polyhedron in R P^{6g−5} [H]. We do not know whether this embedding extends to an embedding of 𝒯_{g,0} into R P^{6g−5} .

2. Triangulations and laminations on surfaces with cusps

In this section we construct a homeomorphism of the space of measured laminations on a surface S ∈ 𝒯_{g,m} with m ≥ 1 cusps onto the boundary of a convex cone in R^{6g−5+2m}.

First we look briefly at closed surfaces. A signed geodesic current on a closed surface S of genus g ≥ 2 is a locally finite signed Borel-measure on the space of unoriented geodesics in the hyperbolic plane H^{2} which is invariant under the action of the fundamental group π_{1}(S) of S. The space 𝒮 𝒞 of signed geodesic currents for S, equipped with the weak^{∗} -topology, is a topological vector space which only depends on the topological type of S. It contains the space 𝒞 of geodesic currents which consists of all nonnegative elements of 𝒮 𝒞 as a closed subcone.

There is a bilinear form i on 𝒮 𝒞 , the so called intersection form, whose restriction to 𝒞 is continuous with respect to the weak^{∗} -topology [B], but it is not continuous globally as a form on 𝒮 𝒞 . The subset ℒ of 𝒞 of all geodesic currents μ ∈ 𝒞 with vanishing self-intersection i(μ, μ) = 0 is the closed cone of measured geodesic laminations and is homeomorphic to R^{6g−6} [FLP]. However, ℒ is not contained in any finite-dimensional linear subspace of 𝒮 𝒞 . The projectivization 𝒫ℒ of the space of nonzero measured geodesic laminations defines a compactification of the Teichmüller space 𝒯_{g,0} which is called the Thurston boundary. Every closed geodesic ψ on S can naturally be viewed as a geodesic current and hence via μ → i(ψ, μ) it defines a linear functional on 𝒮 𝒞 whose restriction to 𝒞 and hence to ℒ is continuous.

We can also consider the space ℒ of measured geodesic laminations on hyperbolic surfaces with cusps. By definition, a measured geodesic lamination for such a surface S with m ≥ 1 cusps is a compact subset of S which is foliated by geodesics and equipped with a transverse invariant measure.

Now let m ≥ 1 and let S ∈ 𝒯_{g,m}. Fix one of the cusps of S and denote it by O. Choose 6g − 5 + 2m simple mutually disjoint geodesics η̃_{1}, …, η̃_{6g−5+2m} on S whose two ends go into the cusp O and which decompose S into 4g − 3 + m ideal triangles and m − 1 once-punctured discs (see [S3]).

If ψ is any closed geodesic on S then ψ is contained in a compact subset of S and hence it intersects each of the geodesics η̃_{i} transversely in a finite number of points. We denote by i(ψ, η̃_{i}) the number of intersections of ψ with η̃_{i}. Since measured laminations on S have compact support, intersection of closed geodesics with one of the curves η̃_{i} extends to a continuous convex-linear functional i( · , η̃_{i}) on the space ℒ .

The intersection of each closed geodesic ψ on S with one of the ideal triangles T cut out by the geodesics η̃_{i} consists of a finite number of simple arcs. Each of these arcs has its endpoints on two different sides of T. In other words, the number of intersections of ψ with a fixed side of T is not smaller than the sum of the number of intersections with the two other sides. In particular, the 6g − 5 + 2m-tuple of intersection numbers of ψ with the geodesics η̃_{i} is contained in the intersection of a collection of 10g − 8 + 3m halfspaces bounded by a linear hyperplane through the origin. The boundary 𝒜 ⊂ R^{6g−5+2m} of this convex cone equals the set of all 6g − 5 + 2m-tuples (a_{1}, …, a_{6g−5+2m}) of nonnegative real numbers with the following properties:

(1) a_{i} ≤ a_{j} + a_{k} if the geodesics η̃_{i}, η̃_{j}, η̃_{k} are the sides of an ideal triangle in S.

(2) There is at least one ideal triangle in S with sides η̃_{i}, η̃_{j}, η̃_{k} and such that a_{i} = a_{j} + a_{k}.

In particular, 𝒜 is a cone with vertex at the origin over the boundary ∂P of a convex polyhedron P in the unit sphere S^{6g−6+2m} ⊂ R^{6g−5+2m}. Since the boundary of a convex polyhedron in S^{6g−6+2m} is homeomorphic to a sphere of dimension 6g − 7 + 2m, our set 𝒜 is homeomorphic to R^{6g−6+2m}.

We summarize our discussion in the following lemma.

Lemma 2.1. 𝒜 is a cone with vertex at the origin over the boundary of a convex finite-sided polyhedron in the sphere S^{6g−6+2m}. In particular, 𝒜 is homeomorphic to R^{6g−6+2m}.

Lemma 2.2. The map μ ∈ ℒ → (i(η̃_{1}, μ), …, i(η̃_{6g−5+2m}, μ)) ∈ R^{6g−5+2m} is a homeomorphism of ℒ onto 𝒜 .

Proof. For μ ∈ ℒ write Φ(μ) = (i(η̃_{1}, μ), …, i(η̃_{6g−5+2m}, μ)). We show first that the map Φ is injective. By continuity, for this it suffices to show that every simple closed geodesic multicurve γ (i.e. a finite union of pairwise disjoint simple closed geodesics, possibly multiple covered) is determined by Φ(γ).

By assumption, the arcs η̃_{i} define a decomposition of S into 4g − 3 + m ideal triangles with vertices at the cusp O and m − 1 punctured discs. Each arc is either the common side of exactly two triangles or it is the common side of one triangle and one once-punctured disc. Let γ be a simple closed geodesic multicurve on S and let T be a triangle from our triangulation of S with sides β_{1}, β_{2}, β_{3} . Write j_{i} = i(β_{i}, γ) and assume that j_{1} ≥ j_{2} ≥ j_{3} . Since the interior of T is contractible in the compactification of S, the total intersection number j_{1} + j_{2} + j_{3} of γ with the boundary of T is even and hence j_{2} + j_{3} − j_{1} is even as well. Moreover we have j_{1} ≤ j_{2} + j_{3} . Draw \frac{1}{2}(j_{2} + j_{3} − j_{1}) simple arcs connecting the sides β_{2} and β_{3} , j_{2} − \frac{1}{2}(j_{2} + j_{3} − j_{1}) simple arcs connecting the sides β_{1} and β_{2} , j_{3} − \frac{1}{2}(j_{2} + j_{3} − j_{1}) simple arcs connecting the sides β_{1} and β_{3} in such a way that all these arcs are disjoint. The configuration of these arcs in T is uniquely determined up to homotopy by j_{1}, j_{2}, j_{3} .

If η̃_{i} is the boundary of a once-punctured disc D then each connected component of γ ∩ D has its two endpoints on η̃_{i} and therefore i(γ, η̃_{i}) = 2k for some k ≥ 0. Draw k simple arcs in D with endpoints on η̃_{i} and the additional property that each of these arcs separates the puncture in the interior of D from the puncture on the boundary. Once again, this configuration is determined uniquely up to homotopy by the number 2k.

The thus constructed arcs in the 4g−3+m triangles and m−1 punctured discs of our triangulation can be connected in a unique way to a simple closed multicurve ψ on S. This multicurve is determined up to isotopy by the intersection numbers with the geodesics η̃_{1}, …, η̃_{6g−5+2m}.

To show that ψ is isotopic to γ we just have to verify that each connected component of the intersection of γ with a punctured disc D separates the puncture in the interior from the puncture on the boundary. But this follows from the fact that those arcs are the only simple arcs in D which are not freely homotopic relative to the boundary of D to an arc contained in the boundary. Thus γ is uniquely determined by Φ(γ) (compare the discussion in [FLP]) and Φ is injective.

We are left with showing that the image of ℒ under the map Φ equals the cone 𝒜 . We show first that Φℒ is contained in 𝒜 . For this recall from our consideration above that every 6g − 5 + 2m-tuple (b_{1}, …, b_{6g−5+2m}) of even nonnegative integers with the additional property that b_{i} ≤ b_{j} + b_{k} whenever the geodesics η̃_{i}, η̃_{j}, η̃_{k} are the sides of an ideal triangle determines uniquely a (possibly multiple covered) simple closed multicurve (compare [FLP]). However, not every such multicurve is freely homotopic to a simple geodesic multicurve. Namely, a simple closed curve which is parallel to the cusp O corresponds to the 6g − 5 + 2m-tuple (2, …, 2). Since the homotopy class of this curve cannot be represented by a simple closed geodesic, a 6g −5+2m-tuple of equal even positive integers does not occur as the intersection tuple of a geodesic lamination.

Denote by ℬ ⊂ R^{6g−5+2m} the set of all 6g−5+2m-tuples (b_{1}, …, b_{6g−5+2m}) of nonnegative numbers with the additional property that b_{i} ≤ b_{j} + b_{k} if η̃_{i}, η̃_{j}, η̃_{k} are the sides of a triangle in S. If (b_{1}, …, b_{6g−5+2m}) ∈ ℬ is such that for each triangle in S with sides η̃_{i}, η̃_{j}, η̃_{k} the strict inequality b_{i} < b_{j} + b_{k} holds, then there are β > 0 and (a_{1}, …, a_{6g−5+2m}) ∈ 𝒜 such that (b_{1}, …, b_{6g−5+2m}) = (a_{1}, …, a_{6g−5+2m}) + β(2, …, 2). As before, if the numbers a_{i} are even integers, then the vector (a_{1}, …, a_{6g−5+2m}) defines a unique simple closed multicurve μ in S. Let γ be a simple closed curve which is parallel to the cusp O and which does not intersect μ. Then μ ∪ βγ is a weighted multicurve whose intersection tuple equals (b_{1}, …, b_{6g−5+2m}). By uniqueness, (b_{1}, …, b_{6g−5+2m}) does not define a measured lamination. From continuity we therefore conclude that the image of Φ is contained in 𝒜 .

To show that Φ maps ℒ onto 𝒜 , recall that a tuple (m_{1}, …, m_{6g−5+2m}) ∈ 𝒜 with even integers as coefficients determines uniquely up to isotopy a simple closed multicurve μ which does not contain any puncture parallel component. In particular, μ is isotopic to a geodesic multicurve γ. Our discussion above (compare also [FLP]) shows that the 6g −5+2m-tuple of intersection numbers for γ with the geodesics η̃_{i} coincides with (m_{1}, …, m_{6g−5+2m}). But this just means that we can define a map from 𝒜 to ℒ which is inverse to Φ. This finishes the proof of the lemma.

As a corollary, we find the following well-known fact [FLP].

Corollary 2.3. The space ℒ of measured geodesic laminations on a surface of genus g ≥ 1 with m ≥ 1 cusps is homeomorphic to an open cell of dimension 6g − 6 + 2m.

We assign now to each of the geodesics η̃_{i} a closed geodesic with two selfintersections as follows. For each i the geodesic η̃_{i} is contained in a unique oncepunctured annulus A ⊂ S with geodesic boundary. Let ψ_{i} be the closed geodesic in A with two self-intersections which intersects the perpendicular ν between the boundary geodesics of A twice and is invariant under reflection along ν. The next lemma shows that measured laminations on a surface of genus g with m ≥ 1 punctures can be parameterized by their intersections with the 6g − 5 + 2m closed geodesics ψ_{1}, …, ψ_{6g−5+2m}.

Lemma 2.4. Let μ ∈ ℒ be a measured geodesic lamination. Then i(μ, ψ_{i}) = 2i(μ, η̃_{i}) for all i; in particular, the map

Ψ: μ → (i(μ, ψ_{1}), …, i(μ, ψ_{6g−5+2m}))

is a homeomorphism of ℒ onto 𝒜 .

Proof. As before, it is enough to show the statement of the lemma for simple closed geodesic multicurves. Consider again the once-punctured annulus A containing η̃_{i} and ψ_{i}. Its boundary consists of simple closed geodesics σ, σ̂. There is no simple closed geodesic contained in the interior of A, so the intersection with A of each simple closed geodesic multicurve is a finite collection of simple arcs with endpoints on the boundary. There are only 3 different free homotopy classes of such simple arcs in A relative to the boundary. For each of these classes, it can be checked explicitly that the number of its intersections with the geodesic ψ_{i} is twice the number of its intersections with η̃_{i}.

3. Length functions for surfaces with cusps

In Section 2 we constructed for every g ≥ 0 and m ≥ 1 such that 2g−2+m ≥ 1 a collection of 6g −5+2m free homotopy classes on a surface of genus g with m cusps which parameterize the space of measured geodesic laminations ℒ via intersection. The projectivization of this parameterization defines a homeomorphism of the Thurston boundary of 𝒯_{g,m} onto the boundary of a finite-sided convex polyhedron P in R P^{6g−6+2m}. The purpose of this section is to show that the length functions of these curves define a homeomorphism of Teichmüller space onto the interior of P. For this we continue to use the assumptions and notations from Section 2.

As in Section 2 we fix one of the punctures of a surface S ∈ 𝒯_{g,m} and call it O. Consider once again a collection of 6g − 5 + 2m simple geodesics η̃_{1}, …, η̃_{6g−5+2m} whose two ends go into the cusp O and which decompose S into 4g − 3 ideal triangles and m−1 once-punctured discs. We call this decomposition the preferred triangulation of our surface S, and we call O the preferred puncture. For each i the length of the geodesic η̃_{i} is infinite; however, we can assign a relative length lη̃_{i}(S) to it as follows. The puncture O of S admits a neighborhood in S which is isometric to a standard cusp C [Bu]. By definition, such a standard cusp can be identified with the cylinder [− log 2, ∞) × S^{1} with the metric dϱ^{2} + e^{−2ϱ} dt^{2} . We define the height of a point in C to be the value of the ϱ-coordinate in this representation.

Let △_{∞} be an ideal triangle in H^{2} . It contains a unique finite equilateral triangle T with vertices on the sides of △_{∞} and which is invariant under all isometries of △_{∞}. For every ideal vertex ξ of △_{∞} there is a unique horocircle H at ξ which passes through two of the vertices of T. Choose a number ϱ_{0} > 0 in such a way that 2πe^{−2ϱ_{0}} is smaller than the length of the intersection of the horocircle H with △_{∞}. Explicit computation shows that this is the case if and only if we have ϱ_{0} ≥ − log(sinh(\frac{1}{2}arccosh(\frac{3}{2}))/π) ∼ 1.838. Every geodesic going into the cusp meets the circles ϱ = const orthogonally and hence, since both ends of the geodesic η̃_{i} go into the cusp, each choice of a height cuts from η̃_{i} a unique compact arc of finite length. Denote by lη̃_{i}(S) the length of the subarc of η̃_{i} which corresponds to the height ϱ_{0} .

Proposition 3.1. The map

Λ: S ∈ 𝒯_{g,m} → (lη̃_{1}(S), …, lη̃_{6g−5+2m}(S)) ∈ R^{6g−5+2m}

is a diffeomorphism of 𝒯_{g,m} onto a hypersurface in R^{6g−5+2m}.

Proof. Consider again an ideal triangle △_{∞} in H^{2} with sides a_{1}, a_{2}, a_{3} . We parameterize the geodesics a_{i} by arc length in such a way that the origin corresponds to the vertex z_{i} of the subtriangle T ⊂ △_{∞} on a_{i} and that the orientations of a_{i} define the boundary orientation of △_{∞}. The distinguished points z_{i}, z_{i+1} lie on a common horocircle through the point a_{i}(∞).

For given numbers l_{1}, l_{2}, l_{3} > 0 define

x_{i} = a_{i}(\frac{1}{2}(−l_{i} + l_{i+1} − l_{i+2})), y_{i} = a_{i}(\frac{1}{2}(l_{i} + l_{i+1} − l_{i+2})).

The distance between x_{i} and y_{i} equals l_{i}, and the points y_{i} and x_{i+1} are contained in a common horocircle at a_{i}(∞). If we identify the sides of △_{∞} with R in the above way, then the assignment which maps (l_{1}, l_{2}, l_{3}) to the triple (x_{1}, x_{2}, x_{3}) of points on the three different sides of △_{∞} is a real analytic diffeomorphism onto its image.

Now let (l_{1}, …, l_{6g−5+2m}) be any 6g − 5 + 2m-tuple of positive numbers. If i, j, k ≤ 6g −5+2m are such that the geodesics η̃_{i}, η̃_{j}, η̃_{k} are the sides of a triangle of our preferred triangulation, then the triple (l_{i}, l_{j}, l_{k}) determines uniquely three arcs of length l_{i}, l_{j}, l_{k} on the three sides of an ideal triangle △_{∞}. Moreover, there is a distinguished point on the boundary geodesic of a once-punctured disc which is just the orthogonal projection of the puncture to the boundary. Each number l ≥ 0 then determines a compact arc of length l on the boundary of a once-punctured disc whose midpoint is this distinguished point.

Each geodesic η̃_{i} is either a common side of exactly two ideal triangles from our triangulation, or a common side of one triangle and a once-punctured disc. Each such pair of sides can be glued with an isometry which identifies the distinguished arcs of length l_{i} and reverses the orientation. The result of these 6g − 5 + 2m glueings is a complete hyperbolic surface with m cusps and a distinguished horocircle going around one of the cusps. The length of this horocircle is the sum of the lengths of all the horocyclic arcs connecting pairs of endpoints of our distinguished boundary segments in each of the triangles and once-punctured discs.

Our construction defines a real analytic map μ of R6g−5+2m_{+} into 𝒯_{g,m} which satisfies μ∘Λ = Id and hence is surjective. A point (l_{1}, …, l_{6g−5+2m}) ∈ R6g−5+2m_{+} belongs to the image of Λ if and only if the total length of the distinguished horocircle in the surface constructed from (l_{1}, …, l_{6g−5+2m}) in the above way equals 2πe^{−2ϱ_{0}} . In particular, the map Λ is a real analytic diffeomorphism of 𝒯_{g,m} onto a smooth hypersurface in R^{6g−5+2m}.

Let Π: R^{6g−5+2m} − {0} → R P^{6g−6+2m} be the canonical projection. The map Π ∘ Λ: 𝒯_{g,m} → R P^{6g−6+2m} is real analytic.

Corollary 3.2. The map Π∘Λ is a diffeomorphism of 𝒯_{g,m} onto the interior of a finite-sided closed convex polyhedron P in R P^{6g−6+2m} which extends to a homeomorphism of 𝒯_{g,m} ∪ ∂𝒯_{g,m} onto P.

Proof. Let again △_{∞} be an ideal triangle in H^{2} and consider a horocyclic arc through one of its ideal vertices ζ which passes through the two distinguished points on the sides adjacent to ζ. By the choice of ϱ_{0} , the length of this arc is bigger than 2πe^{−2ϱ_{0}} . It therefore follows from the construction in the proof of Proposition 3.1 that for every (l_{1}, …, l_{6g−5+2m}) ∈ Λ𝒯_{g,m} and for every triple i, j, k which defines a triangle of our triangulation we have l_{i} < l_{j} + l_{k}.

Denote by a_{1}, a_{2}, a_{3} the sides of the ideal triangle △_{∞} and let (l_{1}, l_{2}, l_{3}) be a triple of positive reals such that l_{i} < l_{i+1} + l_{i+2} for all i (indices are taken mod 3). In the proof of Proposition 3.1 we constructed from this triple for each i a subsegment α_{i} of length l_{i} of the side a_{i} of △_{∞}. For ε > 0 denote by α̃_{i} the subsegment of a_{i} which is induced by the triple ((1 + ε)l_{1}, (1 + ε)l_{2}, (1 + ε)l_{3}). Since l_{i} < l_{i+1} + l_{i+2} , our explicit construction of the segments α_{i}, α̃_{i} shows that α_{i} is a subarc of α̃_{i}. But this means that the length of each of the three horocyclic arcs in △_{∞} connecting pairs of endpoints of the segments α̃_{i} as above is strictly shorter than the length of the corresponding horocyclic arc connecting endpoints of the segments α_{i}. Similarly, the horocyclic arc connecting points on the boundary geodesic of a once-punctured disc which is determined by the length (1 + ε)l is shorter than the arc determined by the length l.

Thus replacing a tuple (l_{1}, …, l_{6g−5+2m}) in our construction above by a multiple with factor 1 + ε > 1 results in decreasing the length of our distinguished horocircle. In particular, Π ∘ Λ is injective. The proposition now follows from this and Lemma 2.2.

In Section 2 we associated to each of the geodesics η̃_{i} on a surface S ∈ 𝒯_{g,m} a closed geodesic ψ_{i} (i = 1, …, 6g − 5 + 2m) on S. The “length” lη̃_{i}(S) of η̃_{i} on S can be computed from the length l_{ψ_{i}}(S) of ψ_{i} as follows. In the hyperbolic plane H^{2} draw a geodesic segment ν of length l_{ψ_{i}}(S)/4. Let ζ_{1}, ζ_{2} be the orthogonals of ν through the endpoints of ν. There is a unique geodesic line ζ_{3} in H^{2} with one endpoint at ζ_{1}(∞) which intersects the geodesic ζ_{2} orthogonally in a point x. Through each point of ζ_{3} passes a horocircle at ζ_{3}(∞) = ζ_{1}(∞) which intersects the geodesic ζ_{1} . The length of the subarc of this horocircle with endpoints on ζ_{1} and ζ_{3} decreases exponentially along ζ_{3} . Thus there is a unique point y on ζ_{3} such that this length equals πe^{−2ϱ_{0}} . Then lη̃_{1}(S)/2 is the oriented distance between x and y. This observation is summarized in the following lemma.

Lemma 3.3. There is a strictly increasing real analytic function φ: [0, ∞) → [0, ∞) such that for every i ≤ 6g − 5 + 2m and every surface S ∈ 𝒯_{g,m} we have lη̃_{i}(S) = φ(l_{ψ_{i}}(S)). In particular, the map

Ψ: S ∈ 𝒯_{g,m} → (l_{ψ_{1}}(S), …, l_{ψ_{6g−5+2m}}(S)) ∈ R^{6g−5+2m}

is a diffeomorphism onto its image.

The fact that a surface S ∈ 𝒯_{g,m} is uniquely determined by the lengths of the geodesics ψ_{i} was earlier observed by Seppälä and Sorvali [SS2] (see also [S3]).

Recall from Section 2 that the image of the space 𝒫ℒ of projective measured laminations under the map [μ] → [i(ψ_{1}, μ), …, i(ψ_{6g−5+2m}, μ)] is the boundary ∂P of a compact convex polyhedron P in R P^{6g−6+2m} with finitely many sides.

Lemma 3.4. The image of 𝒯_{g,m} under the map Π ∘ Ψ equals the interior of the convex polyhedron P.

Proof. It follows from Lemma 2.4 that the map Π∘Ψ extends continuously to a homeomorphism of ∂𝒯_{g,m} onto the boundary ∂P of the polyhedron P. Since 𝒯_{g,m} ∪ ∂𝒯_{g,m} is homeomorphic to a closed ball it is therefore enough to show that the image of 𝒯_{g,m} under Π ∘ Ψ is contained in the interior of P.

For j ∈ {1, …, 6g − 5 + 2m} and a measured lamination μ on a surface of genus g with m punctures define α_{j}(μ) = i(ψ_{j}, μ). Then α_{j} is a linear functional on the space ℒ of measured laminations. Each side B of codimension 1 of our polyhedron P is either the projection of a hyperplane defined by the equation α_{j} = 0 for some j ∈ {1, …, 6g − 5 + 2m} or there is a triangle of our preferred triangulation with sides η̃_{i}, η̃_{j}, η̃_{k} and such that B is the intersection with P of the projection of the linear hyperplane defined by α_{i} + α_{j} − α_{k} = 0. Thus it is enough to show that whenever η̃_{i}, η̃_{j}, η̃_{k} are the sides of a triangle of our preferred triangulation, then l_{ψ_{i}}(S) + l_{ψ_{j}}(S) − l_{ψ_{k}}(S) > 0 for every S ∈ 𝒯_{g,m}.

By Lemma 3.3 there is a real analytic monotonously increasing function f: (0, ∞) → (0, ∞) such that lη̃_{i}(S)/2 = f(l_{ψ_{i}}(S)/4) for all i ≤ 6g − 5 + 2m and all S ∈ 𝒯_{g,m}. The function f can be computed explicitly as follows (see [S3]). For some small ε > 0 replace our surface S with cusps by a surface S_{ε} with geodesic boundary and such that the length of each boundary component equals ε. Define the length lη̃_{i}(S_{ε}) as the length of the perpendicular to the boundary component of S_{ε} which corresponds to the preferred cusp O of S and whose free homotopy class relative to the boundary corresponds to η̃_{i}. The closed geodesic ψ_{i} on S_{ε} and its length l_{ψ_{i}}(S_{ε}) is also defined. Hyperbolic trigonometry shows [S3] that

lη̃_{i}(S_{ε})/2 = arsinh (sinh(ε/2)^{−1} cosh(l_{ψ_{i}}(S_{ε})/4)).

By definition of our functions lη̃_{i} there is a sequence of constants a_{ε} → ∞ such that lη̃_{i}(S_{ε}) − a_{ε} → lη̃_{i}(S) for all S ∈ 𝒯_{g,m}; moreover, we have l_{ψ_{i}}(S_{ε}) → l_{ψ_{i}}(S). Since arsinh (t)−log(t)+log(2) → 0 (t → ∞) we conclude that there is a constant c ∈ R such that

lη̃_{i}(S)/2 = log cosh(l_{ψ_{i}}(S)/4) + c.

It follows from our explicit construction that we may adjust our normalization for the definition of the functions lη̃_{i} (i.e. the choice of the height ϱ_{0} ) in such a way that c = 0. Then we have f(t) = log cosh(t), or equivalently, l_{ψ_{i}}(S)/4 = arcosh e^{lη̃_{i} (S)/2} for all S ∈ 𝒯_{g,m}.

Now lη̃_{i}(S) + lη̃_{j}(S) − lη̃_{k}(S) > 0 for every S ∈ 𝒯_{g,m} and all i, j, k such that the geodesics η̃_{i}, η̃_{j}, η̃_{k} form a triangle of the preferred triangulation. Since the function t → arcosh (e^{t}) is a strictly increasing concave diffeomorphism of the half-line [0, ∞) we conclude that also l_{ψ_{i}}(S) + l_{ψ_{j}}(S) − l_{ψ_{k}}(S) > 0. This shows the lemma.

We can use Lemma 3.3 and Lemma 3.4 to complete the proof of our theorem from the introduction.

Proposition 3.5. The map Π ∘ Ψ: 𝒯_{g,m} → R P^{6g−6+2m} is a diffeomorphism of 𝒯_{g,0} onto the interior of a finite-sided convex polyhedron P in R P^{6g−6+2m} which extends to a homeomorphism of 𝒯_{g,m} ∪ ∂𝒯_{g,m}.

Proof. Since Ψ is a diffeomorphism of 𝒯_{g,m} onto a smooth hypersurface in R^{6g−5+2m} it is enough to show that for every S ∈ 𝒯_{g,m} the line through Ψ(S) and the origin is not tangent to the hypersurface Ψ(𝒯_{g,m}) at Ψ(S).

For this we assume to the contrary that there is a surface S ∈ 𝒯_{g,m} and a smooth curve c(t) ⊂ 𝒯_{g,m} through c(0) = S such that

\frac{d}{dt}Ψ(c(t)) |_{t=0}= Ψ(c(0)).

Hyperbolic trigonometry shows that the lengths of the geodesics ψ_{i} are bounded from below by 2 arsinh (4π) ∼ 6.45. Recall from the proof of Lemma 3.4 that for a suitable choice of a height at the cusp O we have lη̃_{i}(M)/2 = log cosh(l_{ψ_{i}}(M)/4) for all i and all M ∈ 𝒯_{g,m} and therefore

\frac{d}{dt}lη̃_{i}(c(t))/2 |_{t=0}= l_{ψ_{i}}(S) tanh(l_{ψ_{i}}(S)/4)/4.

The function σ: s → log cosh(s)+log(2)−s tanh(s) is monotonously decreasing with s and tends to 0 as s → ∞. Its value at 1.5 is smaller than (log 2)/3. By the above, for s = l_{ψ_{i}}(S)/4 we have

0 < σ(s) = \frac{1}{2}lη̃_{i}(S) + log 2 − \frac{d}{dt}(\frac{1}{2}lη̃_{i}(c(t))) |_{t=0}< \frac{log 2}{3}

and, in particular,

\frac{1}{2}lη̃_{i}(S) + \frac{2}{3} log 2 < \frac{d}{dt}lη̃_{i}(c(t)) |_{t=0}< \frac{1}{2}lη̃_{i}(S) + log 2.

Lemma 3.4 and its proof show that for every triangle of our preferred triangulation with sides η̃_{i}, η̃_{j}, η̃_{k} we have lη̃_{i}(S) < lη̃_{j}(S) + lη̃_{k}(S). From this and the above we conclude that

\frac{d}{dt}lη̃_{i}(c(t)) |_{t=0} +\frac{d}{dt}lη̃_{j}(c(t)) |_{t=0} −\frac{d}{dt}lη̃_{k}(c(t)) |_{t=0}> 0.

Our explicit construction in the proof of Lemma 3.1 then shows that the differential at 0 of the length of the distinguished horocircle in the surface c(t) is negative. This contradicts the definition of the functions lη̃_{i} and therefore our hypersurface Ψ(𝒯_{g,m}) is nowhere tangent to the lines through the origin. This shows the proposition.

4. Length functions on closed surfaces

The purpose of this section is to construct a simple parameterization of 𝒯_{g,0} by 6g − 5 geodesic length functions and hence to give a new and simpler proof of the result of Schmutz Schaller [S1].

For this let S ∈ 𝒯_{g,0} be a closed hyperbolic surface and let ν_{0} be a simple closed separating geodesic on S such that after cutting S open along ν_{0} we obtain a bordered torus T_{0} and a bordered surface S_{0} of genus g − 1. Let ν_{1}, ν_{2}, ν_{3} be simple closed geodesics on T_{0} which mutually intersect in a single point. If we cut S open along ν_{1} then we obtain a connected surface S_{1} of genus g − 1 with two boundary circles. Denote one of the boundary circles by O.

We can use our definition of the geodesic arcs η̃_{i} from Lemma 2.2 also for surfaces with geodesic boundary by requiring that these arcs meet the boundary circle O perpendicularly at both of their endpoints. This then defines a collection of 6g−7 simple geodesic arcs η̃_{1}, …, η̃_{6g−7} on S_{1} which decompose S_{1} into 4g−5 right-angled hexagons and one annulus A with piecewise geodesic boundary. One boundary component of A is the boundary circle of S_{1} different from O. We choose the numbering of our geodesic arcs in such a way that the second boundary component of A contains the geodesic η̃_{6g−7} as a subarc.

For each i ≤ 6g − 8 the geodesic η̃_{i} is contained in a unique pair of pants P ⊂ S_{1} whose geodesic boundary consists of the circle O and two additional simple closed geodesics σ, σ̂. Let ψ_{i} be the closed geodesic in P with two selfintersections which intersects the perpendicular ν of the geodesics σ, σ̂ twice and is invariant under reflection along ν. We view ψ_{i} as a geodesic on S. Write, moreover, ψ_{6g−8+i} = ν_{i} ⊂ T_{0} ⊂ S.

We can now show that the length functions of the family ψ_{1}, …, ψ_{6g−5} of 6g − 5 closed geodesics define an embedding of 𝒯_{g,0} into R P^{6g−5} .

Proposition 4.1. The map

Ψ_{0}: S ∈ 𝒯_{g,0} → (l_{ψ_{1}}(S), …, l_{ψ_{6g−5}}(S)) ∈ R^{6g−5}

is a diffeomorphism onto its image.

Proof. Consider the subtorus T_{0} of the closed hyperbolic surface S of genus g. The simple closed geodesics ν_{i} = ψ_{6g−8+i} ⊂ T_{0} (i = 1, 2, 3) mutually intersect in a single point. Seppälä and Sorvali ([SS1], compare also [SS5] and [S1]) showed that the length functions of the geodesics ν_{1}, ν_{2}, ν_{3} determine the hyperbolic structure of T_{0} and the length of its boundary geodesic.

Let again S_{1} be the subsurface of S of genus g − 1 with two boundary circles which we obtain by cutting S along the geodesic ν_{1} = ψ_{6g−7} ⊂ T_{0} . Recall from Section 2 the definition of the geodesic arc η̃_{6g−7} in S_{1} ; it is contained in the torus T_{0} and therefore its length is determined by the lengths of the geodesics ν_{i}.

As above let η̃_{j} (j ≤ 6g − 8) be the simple geodesic arcs in S_{1} which decompose S_{1} into 4g − 5 right-angled hexagons and one annulus and which we used to define our geodesics ψ_{j}. Hyperbolic trigonometry shows that

cosh(l_{ψ_{i}}(S)/4) = sinh(lη̃_{i}(S)/2) sinh(l_{ν_{1}}(S)/2)

(compare [S3]). In particular, the lengths of the arcs η̃_{j} (1 ≤ j ≤ 6g − 8) are determined by the lengths of the geodesics ψ_{j} and the length of the boundary geodesic ν_{1} = ψ_{6g−7} . By construction, the arcs η̃_{j} (j = 1, …, 6g − 7) determine the hyperbolic structure of a collection of 4g − 5 right-angled hexagons which can be glued in a unique way to form the complement Ŝ_{1} of an annulus A in the surface S_{1} . The boundary of A consists of one closed geodesic and one rightangled geodesic bigon. The hyperbolic structure of the annulus is completely determined by the length l_{ν_{1}}(S) of the boundary circle and the length of the arc η̃_{6g−7} . Moreover, there is a unique way to glue A to our subsurface Ŝ_{1} of S_{1} which is composed of our right-angled hexagons and such that we obtain a smooth hyperbolic surface with two boundary components. But this just means that the hyperbolic structure of S_{1} is determined.

The surface S is obtained from S_{1} by glueing the two boundary geodesics with a suitable twist. Since the boundary of S_{1} is contained in the torus T_{0} , the twist parameter for the glueing is determined by the hyperbolic structure on T_{0} . Thus the lengths of the geodesics ψ_{j} determine the hyperbolic structure on S.

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88 Ursula Hamenstädt

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Received 6 November 2001