Parametrizations of Teichmüller space and its Thurston boundary

Ursula Hamenstädt ^{∗} Mathematisches Institut der Universität Bonn e-mail: ursula@math.uni-bonn.de

Abstract

For g ≥ 2 let T_{g} be the Teichmüller space of hyperbolic metrics on a closed surface of genus g, and let ∂T_{g} be its Thurston boundary. Using intersection with 6g − 5 simple closed geodesics, we construct an embedding of ∂T_{g} into the real projective space ℝP^{6g−6}. ^{1}

1 Introduction

The easiest way to describe all hyperbolic metrics on a closed surface S_{0} of genus g ≥ 2 is to look at the Teichmüller space T_{g} of marked hyperbolic metrics on S_{0}. This Teichmüller space 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} is homeomorphic to an open cell of dimension 6g − 6.

Let γ be a closed curve on the base surface S_{0} which is not null-homotopic. For every (f, h) ∈ T_{g}, the curve f(γ) is freely homotopic to a unique closed geodesic with respect to the hyperbolic metric h. The length ℓ_{γ}(f, h) of this geodesic defines a smooth (in fact real analytic) function on T_{g} which we call the length function of γ. Thus for a number k > 0 and an ordered k-tuple of fixed nontrivial free homotopy classes [γ_{1}], …, [γ_{k}] on S_{0} we obtain a real analytic map of T_{g} into the k-dimensional euclidean space ℝ^{k} by assigning to (f, h) the nonzero vector (ℓ_{γ_{1}}(f, h), …, ℓ_{γ_{k}}(f, h)) ∈ ℝ^{k}. We call such a map geometric, and special geometric if it is defined by pairwise distinct free homotopy classes of simple closed curves, i.e. curves without self-intersections.

∗Partially supported by the Sonderforschungsbereich 256.

1AMS subject classification: 30F60

There are finite collections of simple closed curves on S_{0} for which the corresponding geometric map is an embedding of Teichmüller space (see [B]). One can obtain rather easily such an embedding with a collection of 9g − 9 simple closed curves. On the other hand it has been known for a long time that there are no such embeddings by 6g − 6 length functions (see the discussion in [S1] and the references there). The question on the minimal number of curves needed to obtain an embedding was answered by Schmutz. He showed the following theorem [S1].

Theorem A: For every g ≥ 2 the Teichmüller space T_{g} can be parametrized by 6g − 5 length functions of simple closed geodesics.

Below we give a proof of Theorem A which is a modification and simplification of the construction of Schmutz. Another very simple geometric embedding of T_{g} into ℝ^{6g−5} can be found in [H]; however this embedding is not special.

A geodesic current for a closed hyperbolic surface S is a locally finite 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 C of geodesic currents for S only depends on the topological type of S and will be equipped with the weak^{∗}-topology. There is a continuous bilinear form i on C, the so called intersection form, which extends the geometric intersection number between two closed geodesics on S viewed as geodesic currents [Bo]. The subset L of C of all geodesic currents μ with vanishing self-intersection i(μ, μ) = 0 is the space of measured geodesic laminations and is homeomorphic to ℝ^{6g−6} [Bo]. Every weighted simple closed geodesic can naturally be viewed as a geodesic current, and this set of currents is dense in L.

Denote by ∂T_{g} the projectivization of L−{0}. With respect to the natural quotient topology, ∂T_{g} is a compact space. Following Thurston (see [FLP]), we say that a sequence of points {(f_{i}, h_{i})}_{i} ⊂ T_{g} converges in T_{g} ∪ ∂T_{g} to the projective class [μ] of a nontrivial measured lamination μ ∈ L if and only if there are numbers a_{i} > 0 (i > 0) such that for every k-tuple of free homotopy classes of curves γ_{1}, …, γ_{k} on S the sequence {a_{i}(ℓ_{γ_{1}}(f_{i}, h_{i}), …, ℓ_{γ_{k}}(f_{i}, h_{i}))}_{i} converges in ℝ^{k} to (i(γ_{1}, μ), …, i(γ_{k}, μ)).

Thurston showed that this defines a topology on T_{g}∪∂T_{g} which restricts to the usual topologies on T_{g} and ∂T_{g}. With respect to this topology the space T_{g} ∪ ∂T_{g} is homeomorphic to a closed ball of dimension 6g − 6 (see [FLP]). In particular, the space of projective measured laminations is homeomorphic to a sphere of dimension 6g − 7. It is called the Thurston boundary of T_{g}.

Recall that a collection ψ_{1}, …, ψ_{k} of simple closed geodesics on S fills up S if every geodesic on S intersects one of the curves ψ_{i} transversely. This is equivalent to saying that the complement of {ψ_{1}, …, ψ_{k}} in S consists of a finite collection of connected simply connected regions. If the curves ψ_{1}, …, ψ_{k} fill up S then for every geodesic lamination μ the vector (i(ψ_{1}, μ), …, i(ψ_{k}, μ)) is non-zero and therefore it defines a point in the real projective space ℝP^{k−1}. In particular, the projectivization of the geometric map defined by ψ_{1}, …, ψ_{k} is a map of T_{g} into ℝP^{k−1} which extends continuously to the Thurston boundary. We call such a map projective geometric, and special projective geometric if it is defined by simple closed curves.

As before, it is well known (see [FLP]) that there are special projective geometric maps which embed T_{g}∪∂T_{g} into a real projective space of dimension not bigger than 9g − 9. More is known for surfaces with cusps where our above discussion is also valid. Namely, the Teichmüller space of all marked complete finite volume hyperbolic structures on a noncompact punctured surface and its Thurston boundary admit projective geometric embeddings onto a compact finite sided polyhedron in a real projective space of minimal dimension [H]. However these embeddings are not special.

For closed surfaces of genus g the existence of projective geometric embeddings of T_{g} ∪∂T_{g} into ℝP^{6g−6} is an open problem. The only partial result is due to Schmutz [S2] who constructed a special geometric embedding of T_{2} into ℝP^{6}. However this embedding does not extend to an embedding of the Thurston boundary. We show below that the projectivization of the geometric embedding of T_{g} into ℝP^{6g−5} from Theorem A also does not extend to an injective map on the Thurston boundary.

This leads us to the simpler question whether the Thurston boundary ∂T_{g} of T_{g} can be parametrized by intersections with 6g − 5 simple closed curves. This question can be answered affirmatively. We show.

Theorem B: For every g ≥ 2 there is a special projective geometric embedding of ∂T_{g} onto the boundary of a finite sided polyhedron in ℝP^{6g−6}.

We do not know whether our curve system which we construct in Theorem B defines an embedding of T_{g} into ℝP^{6g−6}. We conjecture however that this is true and that the image of this embedding equals the interior of our polyhedron.

2 A parametrization of Teichmüller space

In this section we describe a parametrization of Teichmüller space by 6g − 5 length functions. Our construction is a modification of the construction of Schmutz [S1].

First, recall that a pair of pants is a surface with boundary which is homeomorphic to a sphere with 3 holes. Let P_{1}, …, P_{2g−2} be a collection of 2g − 2 pairs of pants. For i ≤ g − 1 label the boundary circles of P_{2i−1} by b_{2i−1}, b_{2i}, d_{i}. For i ≤ g − 2 label the boundary circles of P_{2i} by b_{2i}, b_{2i+1}, d_{i}. The boundary circles of P_{2g−2} are labeled by b_{2g−2}, b_{1}, d_{g−1}.

Glue the pairs of pants along the boundary circles with the same label. We obtain a closed surface S of genus g with a decomposition into 2g−2 pairs of pants which is given by the circles b_{1}, …, b_{2g−2} and the circles d_{1}, …, d_{g−1}.

For every hyperbolic structure on S the distinguished circles are freely homotopic to simple closed geodesics which decompose S into 2g − 2 pairs of pants with hyperbolic structures and geodesic boundaries. In the sequel we denote these geodesics by the same letters b_{1}, …, b_{2g−2}, d_{1}, …, d_{g−1}.

Recall that a Dehn twist along the simple closed curve b_{i} is a diffeomorphism of the surface S which leaves the complement of an embedded cylinder C in S with core-curve b_{i} pointwise fixed. It maps a simple arc c in C with endpoints on the different boundary components of C to an arc with the same endpoints which goes around the cylinder one more time.

Let a_{0} be a simple closed geodesic which intersects each of the curves b_{i} transversely in a single point and does not intersect any of the curves d_{j}. Denote by a_{i} the geodesic obtained from a_{0} by a Dehn twist along b_{i}. The curves a_{0}, a_{i}, b_{i} then mutually intersect transversely in a unique point.

Similarly, define e_{0} to be a simple closed geodesic which intersects each of the curves b_{2j−1}, d_{j} in a single point and does not intersect any of the curves b_{2j}. Let e_{j} be the image of e_{0} under a Dehn twist along d_{j}.

Notice that all our above closed geodesics are simple and non-separating and mutually intersect in at most one point. We have (compare [S1]).

Proposition 2.1: The special geometric map defined by the curves

b_{1}, …, b_{2g−2}, a_{0}, …, a_{2g−2}, d_{1}, …, d_{g−2}, e_{0}, …, e_{g−1}

is an embedding of T_{g} into ℝP^{6g−5}.

Proof: Denote by F our above family of geodesics. We have to show that the hyperbolic structure on our surface S is determined by the lengths of the curves from the collection F. For this it is sufficent to show that the lengths of the boundary circles b_{1}, …, b_{2g−2}, d_{1}, …, d_{g−1} of our pants decomposition of S and the twist parameters which determine how these pants are glued along their boundaries are determined by the lengths of the curves from F.

Cut S open along the curves d_{1}, …, d_{g−1}. We obtain a bordered torus with 2g − 2 boundary circles. Each of these circles is a copy of one of the curves d_{1}, …, d_{g−1}. Following Schmutz (Proposition 1 and the discussion on p.283 of [S1]), the hyperbolic structure of this torus and the common length of the two boundary circles which correspond to the curve d_{g−1} are determined by the lengths of the geodesics d_{1}, …, d_{g−2} and the lengths of the inner geodesics b_{1}, …, b_{2g−2}, a_{0}, …, a_{2g−2} (ses also the discussion below). In particular, for every i ∈ {1, …, 2g − 2} the twist parameter for the glueing of P_{i}, P_{i+1} along b_{i} is determined.

Consider the bordered torus T with 2g + 2 boundary circles which we obtain by cutting S open along the curves b_{2i} (i = 1, …, g − 1). We have to show that the hyperbolic structure on T is determined by the lengths of the inner geodesics e_{0}, …, e_{g−1}, d_{1}, …, d_{g−1} and the hyperbolic structures of the X-pieces X_{i} which we obtain by glueing P_{2i} and P_{2i+1} along the geodesic b_{2i+1}.

For this recall that e_{i} is the image of e_{0} under a full Dehn twist along d_{i}. In particular, the geodesics e_{0}, e_{i}, d_{i} intersect mutually in a single point. The angles between them at these intersection points are determined by their lengths on S [S1].

We argue now as in [S1] and show that the twist parameters for glueing along d_{i} are determined. Namely, choose a lift of e_{0} to the universal cover H^{2}. We obtain a geodesic segment α of a given length ℓ = ℓ_{e_{0}}.

Since the hyperbolic structure on the subsurface X_{i} of T is determined, we know the length of the shortest arc c_{i} in X_{i} connecting the geodesics d_{i} and d_{i+1} and which is freely homotopic to the arc e_{0} ∩ X_{i} relative to the boundary curves d_{i}, d_{i+1}. The subsection α_{i} of α which is the lift of e_{0} ∩ X_{i} forms together with segments from lifts of d_{i}, d_{i+1} and a lift c̃_{i} of c_{i} a geodesic quadrangle Q_{i} with two right angles at the endpoints of c̃_{i}. The length of the side c̃_{i} of Q_{i} and all the angles of Q_{i} are known. Hyperbolic trigonometry shows that Q_{i} is completely determined by these data. Thus the length of the arc α_{i} is also known.

Now the twist parameter for the glueing of X_{i}, X_{i+1} along d_{i+1} can be computed from the signed difference between the lengths of the sides of Q_{i} and Q_{i+1} which are contained in our lift of d_{i+1}. This then shows that the twist parameters for glueing along the geodesics d_{1}, …, d_{g−1} are determined by the lengths of the curves a_{i}, b_{j}, d_{k}, e_{ℓ}. q.e.d.

We do not know whether the projective geometric map of T_{g} into ℝP^{6g−6} defined by our above curves is an embedding. However it is easy to see that its extension to the Thurston boundary is not injective.

Lemma 2.2: The map which assigns to a projective measured lamination [μ] the projective class of the 6g − 5-tuple of intersection numbers with our curves

b_{1}, …, b_{2g−2}, a_{0}, …, a_{2g−2}, d_{1}, …, d_{g−1}, e_{0}, …, e_{g−1}

is not injective.

Proof: It is sufficient to find two distinct simple closed geodesics γ_{1}, γ_{2} on S which have the same number of intersections with each of our above curves b_{i}, a_{j}, d_{k}, e_{ℓ}.

For this let c_{0} be the image of a_{0} under a full Dehn twist along b_{2} and a full Dehn twist along b_{4}. Let c_{1} be the image of c_{0} under a full Dehn twist along b_{2}, and let c_{2} be the image of e_{0} under a full Dehn twist along b_{4}. By the definition of our curve system we have i(a_{i}, c_{j}) = i(a_{i}, c_{0}) + 1, i(b_{i}, c_{j}) = i(b_{i}, c_{0}), i(d_{i}, c_{j}) = i(d_{i}, c_{0}), i(e_{i}, c_{j}) = i(e_{i}, c_{0}) for all i and j = 1, 2 from which our lemma follows. q.e.d.

3 A parametrization of the Thurston boundary

Let b_{1}, …, b_{3g−3} be any family of pairwise disjoint simple closed geodesics on a closed hyperbolic surface S of genus g ≥ 2. These curves decompose our surface S into 2g − 2 pairs of pants. By the Dehn-Thurston theorem (see [HP]), a simple closed geodesic γ is uniquely determined by its intersection numbers with the curves b_{j} and its twist parameters at b_{j}. The twist parameter τ_{j}(γ) of γ at b_{j} is an integer which measures the signed number of times our curve winds around b_{j} with respect to some reference arc. Thus if γ is any simple closed geodesic then the image γ′ of γ under a positive Dehn twist along b_{j} satisfies i(γ′, b_{i}) = i(γ, b_{i}) for i = 1, …, 3g − 3, τ_{k}(γ′) = τ_{k}(γ) for k ≠ j and τ_{j}(γ′) = τ_{j}(γ) + 1.

Choose a simple closed curve a_{0} on S which intersects each of the curves b_{i} in at least one point and has vanishing twist parameters. For i = 1, …, 3g − 3 let a_{i} be the image of a_{0} under a full Dehn twist along b_{i}. Moreover, for every j ∈ {1, …, 3g − 3} choose a simple closed curve c_{j} with vanishing twist parameters and such that the (3g − 3, 3g − 3)-matrix A with entry i(b_{i}, c_{j}) in the i-th line and j-th row has maximal rank.

We can obtain such a pants decomposition and corresponding family of curves c_{j} as follows. Choose g−1 separating simple closed geodesics b_{2}, …, b_{g} which cut S into g − 2 tori T_{2}, …, T_{g−1} with two boundary circles and 2 tori T_{1}, T_{g} with one boundary circle. Assume that the geodesic b_{i} separates T_{i−1} from T_{i}. Extend this system of curves to a pants decomposition of S by simple closed geodesics b_{1}, …, b_{3g−3}.

For each i ∈ {2, …, g} choose a simple closed nonseparating geodesic c_{i} which intersects b_{i} in exactly two points and does not intersect b_{j} for j ≠ i. Such a curve is contained in the subsurface T_{i−1} ∪ T_{i} of S. Choose moreover a simple closed geodesic c_{1}, c_{g+1} in T_{1}, T_{g} which intersects the curve from our pants decomposition contained in T_{1}, T_{g} in a single point.

Let i ∈ {2, …, g − 1} and denote by b′, b′′ the geodesics from our pants decomposition which are contained in T_{i}. If we cut T_{i} open along b′′ then we obtain a sphere with 4 holes. The simple closed geodesic b′ separates two of these holes from the other two. Choose a simple closed geodesic c′ which is contained in this sphere and intersects b′ in exactly two points. Similarly we can find a simple closed geodesic c′′ which intersects b′′ in exactly two points and does not intersect b′. In this way we construct successively a family of curves c_{1}, …, c_{3g−3} with the property that with respect to a suitable numbering we have i(c_{i}, b_{i}) = 2 for i = 2, …, g, g + 2, …, 3g − 3, i(c_{1}, b_{1}) = i(c_{g+1}, b_{g}) = 1 and i(b_{i}, c_{j}) = 0 otherwise.

We show.

Proposition 3.1: The special projective geometric map defined by the curves a_{0}, …, a_{3g−3}, c_{1}, …, c_{3g−3} is an embedding of ∂T_{g} into ℝP^{6g−6}.

Proof: Recall that a simple geodesic multicurve is a collection of mutually disjoint simple closed geodesics with nonnegative multiplicities on our surface S. Simple geodesic multicurves are dense in the space of measured geodesic lamination and therefore it is enough to show that a simple geodesic multicurve is determined by its intersections with our 6g − 5 geodesics a_{i}, c_{j}.

For this recall that the curves b_{1}, …, b_{3g−3} define a pants decomposition of S and therefore a simple geodesic multicurve is uniquely determined by its intersections with the curves b_{j} and its twist parameters at b_{j}.

We show first that the intersection numbers i(γ, a_{i}) determine the intersection numbers i(γ, b_{j}) (j = 1, …, 3g − 3) and the signs of all the twist parameters.

Namely, denote by D_{j} the Dehn twist along b_{j} such that a_{j} = D_{j}a_{0}. Recall that D_{j} leaves the complement of a cylinder C_{j} about b_{j} pointwise fixed. Assume that i(γ, b_{j}) ≠ 0 and let m_{j} be the twist parameter of γ at b_{j}. If i(γ, b_{j}) = k then the multicurve γ intersects the cylinder C_{j} in k arcs. Up to homotopy, each of these arcs meets b_{j} in a unique point, and it has |m_{j}|i(b_{j}, a_{0}) intersection points with the multi-arc a_{0} ∩ C_{j} and |m_{j} − 1|i(b_{j}, a_{0}) intersection points with the multi-arc a_{j} ∩ C_{j}. Outside the cylinder C_{j} the number of intersection points between γ and a_{0}, a_{j} coincide. But this just means that i(γ, b_{j}) = |i(γ, a_{0}) − i(γ, a_{j})|/i(a_{0}, b_{j}). Moreover our twist parameter m_{j} is positive if and only if i(γ, a_{j}) < i(γ, a_{0}), and it is nonpositive otherwise.

We are left with showing that the absolute values of the twist parameters are also determined. For this it is enough to show that for any simple geodesic multicurve and any 3g − 3-tuple k_{1}, …, k_{3g−3} of integers the intersection numbers with the geodesics a_{i}, c_{j} of the multicurves γ and D_{1}^{k_{1}}∘⋯∘D_{3g−3}^{k_{3g−3}}γ = γ′ do not all coincide.

Recall that the signs of the twist parameters are determined by the intersections with the curves a_{j}, so we may assume that the twist parameters of γ and γ′ have the same sign. Write q(i) = i(γ, b_{i}) if the twist parameter of γ at b_{i} is positive, and write q(i) = −i(γ, b_{i}) otherwise. Since the twist parameters of the curve c_{j} vanish, for each i the Dehn twist D_{i}^{k_{i}} applied to γ changes the intersection of γ with the multi-arc c_{j} ∩ C_{i} in the cylinder C_{i} with core curve b_{i} by q(i)k_{i}i(b_{i}, c_{j}).

Thus we have i(D_{i}^{k_{i}}γ, c_{j}) = i(γ, c_{j}) + k_{i}i(b_{i}, c_{j})q(i) and consequently our above statement follows if we can show that the system of 3g − 3 equations

∑_{i=1}^{3g−3} k_{i}i(b_{i}, c_{j})q(i) = 0 (j = 1, …, 3g − 3)

only has solutions which satisfy k_{i} = 0 for all i such that q(i) ≠ 0. Let E be the (3g − 3, 3g − 3)-matrix whose j-th line is the vector

e_{j} = (i(b_{1}, c_{j})q(1), …, i(b_{3g−3}, c_{j})q(3g − 3)).

After reordering of our indices we may assume that q(i) ≠ 0 for i ≤ k and q(i) = 0 otherwise. Then the last 3g − 3 − k rows of the matrix E vanish identically. Since we may multiply a row of E with a nonzero number without changing the rank of E, the rank of C coincides with the rank of a (3g − 3, k)-submatrix of our matrix A. By our assumption on A, the rank of this submatrix equals k and therefore the proposition follows. q.e.d.

The next lemma concludes the proof of our theorem B.

Lemma 3.2: The image of ∂T_{g} under our projective geometric embedding equals the boundary of a finite sided polyhedron in ℝP^{6g−6}.

Proof: We show that the image of the space L of measured laminations under the map Φ which assigns to a lamination μ its 6g −5-tuple of intersection numbers with the geodesics a_{i}, c_{j} equals a finite union of 6g − 6-dimensional convex polyhedra in ℝ^{6g−5}, each contained in a linear hyperplane, and which intersect along their boundaries.

Consider again the pants decomposition of S by the curves b_{1}, …, b_{3g−3}. There is a finite number of train tracks in standard position with respect to this pants decomposition. Each of these train tracks T consists of a finite number of branches and 3-valent switches. Every measured lamination is carried by one of these tracks, i.e. there is such a track T such that the lamination is given by a collection of nonnegative weights on the branches of T which satisfy the algebraic switch conditions.

For every given such train track T the set U of laminations which are carried by T and which are defined by positive weights on all branches of T is an open subset of L. Its boundary consists of laminations which are carried by more than one train track in standard position (for all this see [HP]).

For every μ ∈ U the intersection i(μ, b_{i}) is represented by a weight on one of the branches of T and the same is true for the twist parameter τ_{i}(μ) about b_{i}, where the sign of this twist parameter is determined by the train track T. Since the switch conditions are linear equations in the weights, the set U admits a natural convex linear structure in such a way that the intersection functionals i(·, b_{j}) and the twist parameters τ_{i} are linear.

By the proof of Proposition 3.1, for every i there is an integer p_{i} ≥ 1 such that the intersection number of any measured geodesic lamination μ with the curve b_{i} equals |i(a_{0}, μ) − i(a_{i}, μ)|/p_{i}. Similarly, again by the proof of Proposition 3.1, for any fixed choice of signs for the twist parameters τ_{j} at the pants curves b_{j}, the twist parameters τ_{j}(μ) of a lamination μ with twist parameters of these signs are solutions of 3g −3 independent linear equations in the intersection numbers of μ with the curves c_{j}. Therefore there are 6g−5 linear equations in the intersection numbers i(a_{i}, μ), i(c_{j}, μ) which depend on T and determine the intersection numbers i(μ, b_{j}) and the twist parameters τ_{j}(μ).

Thus if (u_{1}, …, u_{6g−5}) ∈ ℝ^{6g−5} spans the kernel of the corresponding linear map, then the restriction of our map Φ to U is a linear map of a convex cone in an affine space of dimension 6g − 6 onto a convex set in the hyperplane

{(x_{1}, …, x_{6g−5}) ∈ ℝ^{6g−5} | ∑_{i=1}^{6g−5} u_{i}x_{i} = 0}.

But this just means that the image of Φ equals the union of finitely many convex cones in hyperplanes which intersect at their boundaries. Since L is homeomorphic to ℝ^{6g−6} this implies that the image of ∂T_{g} under the special geometric map which is the projectivization of Φ equals the boundary of a finite sided polyhedron in ℝP^{6g−6}. This finishes the proof of our lemma. q.e.d.

4 References

[Bo] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139-162.

[B] P. Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston 1992.

[FLP] A. Fathi, F. Laudenbach, V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque 66-67 (1979).

[H] U. Hamenstädt, Length function and parameterizations of Teichmüller space for surfaces with cusps, preprint 2001.

[HP] R. Harer, J. Penner, Combinatorics of train tracks, Princeton University Press, Princeton 1992.

[S1] P. Schmutz, Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen, Comment. Math. Helvetici 68 (1993), 278-288.

[S2] P. Schmutz Schaller, Teichmüller space and fundamental domains of Fuchsian groups, L’Enseignement Mathématique 45 (1999), 169-187.