Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 151–166

FUNDAMENTAL DOMAINS IN TEICHMÜLLER SPACE

John McCarthy and Athanase Papadopoulos

Michigan State University, Mathematics Department East Lansing, MI 48824, U.S.A.; mccarthy@math.msu.edu CNRS et Université Louis Pasteur, Mathématiques 7 rue René Descartes, F-67084 Strasbourg, France; papadopoulos@math.u-strasbg.fr

Abstract. We introduce a natural construction of fundamental domains for actions of subgroups of the mapping class group on Teichmüller space and investigate their properties. These domains are analogous to the classical Dirichlet polyhedra associated to the actions of discrete isometry groups on hyperbolic spaces.

1. Introduction

In the study of the action of the mapping class group on Teichmüller space (or on Thurston’s compactification of Teichmüller space by the sphere of projective measured laminations), one expects that the actions of this group and of its subgroups will exhibit many of the interesting properties of the actions of discrete groups acting by isometries on a hyperbolic space (a simply connected space with a Riemannian metric of constant negative curvature), and its natural compactification by the sphere at infinity. This reason, of course, is parallel to our motivation for studying, in [MP], the dynamics of the actions of subgroups of the mapping class group on Thurston’s sphere.

In this paper, as in [MP], we develop this line of thought. Specifically, we discuss a natural construction of fundamental domains in Teichmüller space, which are analogous to the Dirichlet polyhedra of the setting of hyperbolic manifolds (which we shall call henceforth the classical setting). It is well known that the actions of subgroups of the mapping class groups are properly discontinuous. Indeed, in [MP], we prove a stronger result. Namely, we can define a useful notion of a limit set for the natural extensions of these actions to Thurston’s compactification of Teichmüller space, and the actions are properly discontinuous on the complement of the limit set, up to a set of measure zero on the boundary sphere. Again, this result has a direct analogy in the classical setting [Th] (without the clause concerning the set of measure zero). Our aim here is not to develop the proper discontinuity. Rather, we wish to study these “Dirichlet polyhedra”.

The outline of the paper is as follows.

1991 Mathematics Subject Classification: Primary 30F60; Secondary 53C70.

In Section 2, we establish notation and recall some preliminary facts from the theory of measured foliations on a surface, Teichmüller spaces and mapping class groups, which will be used in the subsequent text.

In Section 3, we give the definition of the term “fundamental domain”, which is in the same spirit of the one used for example by Beardon in his book [Be]. We then define a domain D in Teichmüller space which is associated to the action of a subgroup Σ of the mapping class group of the surface. As in the classical setting, this domain is defined in terms of equidistant loci and their associated halfspaces. It is well known that there are several metrics that one can define on Teichmüller space (e.g. Teichmüller metric, Weil–Petersson metric, Thurston’s stretch metric). Each one of these metrics is natural from some given point of view. The one that we use is the Teichmüller metric, which has been extensively studied. In particular, we shall use a fact established by Earle [Ea] concerning the differentiability of the Teichmüller distance.

In Section 4, we study the equidistant loci and the associated halfspaces. We establish that an equidistant locus is a hypersurface separating Teichmüller space into two contractible halfspaces. In addition, we show that any two distinct equidistant loci intersect transversely.

In Section 5, we study the domains referred to above. We establish the fact that these domains are fundamental domains. Finally, we discuss the extent to which our domains share the properties of the classical Dirichlet polyhedra.

Let us insist on the fact that the definition of the fundamental domains is classical; the point here is that we are studying such a domain in the context of Teichmüller space, which is not a Riemannian manifold.

This work was done while the authors were visiting the Max-Plank-Institut für Mathematik (Germany), which they thank for its support.

2. Preliminaries

Let S be a finite type surface of negative Euler characteristic which is not homeomorphic to a pair of pants (that is, a sphere with three punctures), and let Γ be the mapping class group of S. Σ will denote an arbitrary subgroup of Γ, and Σ^{∗} will denote the subset of nontrivial elements in Σ.

The Teichmüller space 𝒯 of S is the space of conformal structures on S up to conformal automorphisms which are isotopic to the identity. By the uniformization theorem, we may also consider 𝒯 as the space of hyperbolic metrics on S up to isometries which are isotopic to the identity. For simplicity, we only consider conformal structures on S in which the ends of S are conformally isomorphic to punctured discs. From the geometric point of view, this means that we assume that the ends of S are cusps in the corresponding hyperbolic metrics. The mapping class group Γ acts in a natural way on 𝒯 by the pullback construction.

Teichmüller space has a natural topology in which it is homeomorphic to an open ball of dimension 6g − 6 + 2e where g is the genus of S and e is the number of punctures of S. This topology may be defined by the Teichmüller metric ϱ, a complete metric with respect to which the mapping class group acts as a group of isometries. This metric is not Riemannian. Nevertheless, it provides some of the geometry one associates to Riemannian metrics and in fact, it behaves in some respects better than a generic Riemannian metric. We are particularly interested in the following features of ϱ.

First of all, as shown by Kravetz [Kr], (𝒯 , ϱ) is a straight G-space in the sense of Busemann ([Bu], [Ab]). In particular, any two distinct points in 𝒯 , m and n, are joined by a unique geodesic segment (i.e. an isometric image of a Euclidean interval), [m, n], and lie on a unique geodesic line (i.e. an isometric image of R), γ(m, n). This fact is based on Teichmüller’s Theorem and the following result.

Proposition 2.1. Let m, n and p be points in 𝒯 . If ϱ(m, n) + ϱ(n, p) = ϱ(m, p), then n lies on [m, p].

Proof. See [Ab, p. 122] for a proof based on Teichmüller’s theorem.

As a corollary, we have the following well-known fact which will be useful for us:

Proposition 2.2. Let g be a nontrivial isometry of (𝒯 , ϱ). Then the fixed point set of g is nowhere dense.

Proof. The fixed point set F of g is obviously closed. Hence, it suffices to show that F has empty interior. Suppose, on the contrary, that g fixes a nonempty open set U. Let m be a point in U and n be in 𝒯 . By assumption, g fixes a neighborhood of m in 𝒯 and, hence, in γ(m, n). By uniqueness of geodesics, g preserves the line γ(m, n). But an isometry of a line which fixes an open interval is the identity. Hence, g fixes n. Since n was arbitrary, g is the identity. This is the desired contradiction.

We obtain as a corollary the following (also well known) fact which we shall be using:

Corollary 2.3. There exists a point m_{0} in 𝒯 whose stabilizer in Γ is trivial.

Proof. Since Γ is a countable group, this follows immediately from the Baire category theorem for complete metric spaces.

Choose a point m in 𝒯 . Let ϱ_{m} denote distance from m in 𝒯 .

(2.1) ϱ_{m}: 𝒯 → R, x ⟼ ϱ(x, m).

We shall be interested in the variation of this function. In order to describe this variation, we must discuss the cotangent space of 𝒯 at x. We use the notations of [Ea]. Let X be a Riemann surface representing x. The cotangent space to 𝒯 at x is canonically the vector space Q(X) of integrable holomorphic quadratic differentials on X with the norm, ∥φ∥ = ∫ |φ|/2. It is an easy exercise to establish that this norm is strictly convex.

The norm actually defines a continuous function on the cotangent bundle f: ℰ → 𝒯 . Let ℰ_{0} denote the open set of cotangent vectors of length less than one. If φ in ℰ_{0} is a cotangent vector at x, (i.e., a quadratic differential on X), the Teichmüller differential ∥φ∥\overline{φ}/|φ| on X determines a point g(φ) in 𝒯 . Indeed, this construction gives an “exponential” map Ω_{x} from the unit ball Q_{0}(X) in Q(X) to 𝒯 .

Theorem 2.4 (Teichmüller). The map Ω_{x}: Q_{0}(X) → 𝒯 is a homeomorphism which maps rays in Q_{0}(X) to geodesic rays in 𝒯 emitting from x.

Earle improved upon Teichmüller’s result:

Theorem 2.5 ([Ea]). The map Ω: ℰ_{0} → 𝒯 × 𝒯 is a homeomorphism.

Let ℱ: 𝒯 ×𝒯 → ℰ_{0} denote the inverse of Ω. Earle computed the differential of ϱ_{m} in terms of ℱ.

Theorem 2.6 ([Ea]). Let m be a point in 𝒯 . Then ϱ_{m} is a C^{1} function on 𝒯 \ {m} and its differential is the map x ↦ −ℱ (x, y)/∥ℱ (x, y)∥.

Let Q_{1}(X) be the unit sphere in Q(X). If y is a point on a geodesic ray in 𝒯 emitting from x which is the image under Ω_{x} of a ray {tφ : 0 ≤ t < 1} in Q_{0}(X) for φ in Q_{1}(X), we shall say that φ points in the direction of y. From the definition of the map Ω, we can describe the differential at x.

Corollary 2.7. The differential of ϱ_{m} at x is the unique unit norm quadratic differential on X which points in the direction opposite to m.

This description is what one would expect for a Riemannian metric. The gradient of the distance function measured from m ought to be tangent to the field of geodesic rays emitting from m. Earle’s result confirms this intuition.

Let 𝒮 denote the set of isotopy classes of unoriented, connected and homotopically nontrivial simple closed curves on S which are not homotopic to a puncture of S. The geometric intersection function i( , ) on 𝒮 × 𝒮 is defined by the rule: (2.2) i(α, β) = min{cardinality(a ∩ b) | a ∈ α, b ∈ β}. Γ acts on 𝒮 in a natural way. This function is clearly symmetric and Γ-invariant.

The action of Γ on 𝒮 is a faithful action provided that S is not a closed surface of genus two, a torus with one puncture or a sphere with four punctures. In each of these cases, the kernel of the action is a cyclic subgroup of order two. (These are the only nontrivial maps of surfaces of finite type and negative Euler characteristic which preserve every element of 𝒮 .) We say that the involution is hypergeometric. (2.3) If g is an element of Γ which fixes every simple closed curve in 𝒮 , then g

is either the identity or the hypergeometric involution of a closed surface of genus two, a torus with one puncture or a sphere with four punctures.

By associating to each element α of 𝒮 the length l(m, α) of the corresponding geodesic on X, we obtain a length function, which is evidently Γ-invariant: (2.4) l(gm, gα) = l(m, α) for all g ∈ Γ, m ∈ 𝒯 , α ∈ 𝒮 .

We may choose a finite family 𝒜 of simple closed curves in 𝒮 such that the elements of 𝒮 are parametrized by their geometric intersection with 𝒜 : (2.5) If i(β, α) = i(γ, α) for all α ∈ 𝒜 , then β = γ. We shall say that 𝒜 is a coordinate system.

The Γ-invariance of i implies that Γ acts on coordinate systems. The stabilizers of this action are finite.

Proposition 2.8. Let 𝒜 be a coordinate system. The stabilizer of 𝒜 in Γ is a finite group.

Proof. There is, of course, a subgroup Σ of finite index in the stabilizer which fixes every element of 𝒜 . It suffices to show that Σ is finite. Suppose that g is an element of Σ. Then:

(1) i(α, g^{−1}β) = i(gα, β) = i(α, β) for each β ∈ 𝒮 , α ∈ 𝒜 . By the definition of a coordinate system, it follows that g^{−1} fixes every curve β in 𝒮 . By (2.3), g is either the identity or the hypergeometric involution on S (in the special cases). Hence, Σ is of order at most two.

As an immediate consequence, we have: (2.6) Let 𝒜 and ℬ be coordinate systems. There are at most finitely many map-

ping classes taking 𝒜 to ℬ.

Suppose that {g_{n}} is an infinite sequence of distinct mapping classes. Let 𝒜 be a coordinate system. By (2.6), the collection of coordinate systems {g_{n}(𝒜)} must be infinite (though not necessarily distinct). Hence, for some curve α in 𝒜 , {g_{n}(α)} must also be infinite. By the definition of a coordinate system, the geometric intersection of these curves with 𝒜 must be unbounded. Therefore: (2.7) For any infinite sequence of distinct mapping classes {g_{n}}, there exists a pair

of simple closed curves, α and β, such that the sequence {i(g_{n}(α), β)} is unbounded. There is a useful inequality relating geometric intersection and length.

Lemma 2.9 ([FLP]). Let m be a point in 𝒯 . There is a constant C, depending only on m, such that, for every pair, α and β, of simple closed curves in 𝒮 , one has i(α, β) ≤ Cl(m, α)l(m, β).

There is another important inequality, due to Wolpert, relating the length function to the Teichmüller metric.

Theorem 2.10 ([Wo]). Assume m and n are points in 𝒯 and α is in 𝒮 . Then l(n, α) ≤ e^{ϱ(m,n)}l(m, α).

From this inequality and the previous remarks, we may deduce the proper discontinuity of the Γ-action (and, hence, of the Σ-action).

Theorem 2.11. Γ acts properly discontinuously on 𝒯 .

Proof. Suppose that K is a compact subset of 𝒯 . We wish to show that there are only finitely many mapping classes g in Σ such that the intersection g(K)∩K is nonempty. Suppose, on the contrary, that there is an infinite sequence {f_{n}} of distinct mapping classes such that, for every n, f_{n}(K) ∩ K is nonempty.

Since K is compact, it has a bounded diameter d. Let m be a point in K. Since K meets f_{n}(K):

(1) for every n, ϱ(m, f_{n}(m)) ≤ 2d. If we consider the sequence of mapping classes {g_{n}}, where g_{n} = f_{n}^{−1} , we may apply (2.7) to choose simple closed curves, α and β, such that {i(g_{n}(α), β)} is unbounded. By (2.4) and Lemma 2.9:

(2) {l(f_{n}(m), α)} = {l(m, g_{n}(α))} is unbounded. On the other hand, by Theorem 2.10:

(3) l(f_{n}(m), α) ≤ e^{ϱ(m,f_{n}(m))}l(m, α).

Clearly, (1), (2) and (3) give the desired contradiction.

3. Fundamental domains

As in Beardon [Be], we define a fundamental domain for the action of Σ on 𝒯 as follows. A fundamental set for Σ is a subset of 𝒯 which contains exactly one point from each orbit in 𝒯 . A domain in 𝒯 is an open subset in 𝒯 which is homeomorphic to a ball. A fundamental domain for the action of Σ on 𝒯 is a subset D of 𝒯 which satisfies the following three properties: (3.1) D is a domain in 𝒯 ,

(3.2) there is a fundamental set, F, with D ⊂ F ⊂ \overline{D},

(3.3) ∂D is a connected, properly embedded locally flat submanifold of codimension

one in 𝒯 . We say that the fundamental domain D is locally finite if the following property is also satisfied: (3.4) Every compact subset of 𝒯 meets only finitely many images of D under

elements of Σ.

Note: This is not the same definition of a fundamental domain as used in [Ma] or [MP]. The present notion is more precise.

We shall now proceed with the construction of a fundamental domain D for Σ. In the subsequent sections, we shall establish that D is a locally finite fundamental domain for Σ which is, in some sense, a polyhedron. (We shall be more precise below.)

For the remaining discussion, choose a point m_{0} , as in Corollary 2.3, which is not fixed by any element of Σ^{∗}. Our domain shall be defined exactly as in the classical setting. Hence, we shall be comparing distances.

For each g in Σ^{∗}, we have the equidistant locus from m_{0} to gm_{0} : (3.5) L_{g}(m_{0}) = {m ∈ 𝒯 | ϱ(m, m_{0}) = ϱ(m, gm_{0})}. Note that the stabilizer of gm_{0} is also trivial and: (3.6) L_{g}(m_{0}) = L_{g^{−1}}(gm_{0}).

We shall denote the set of points closer to m_{0} as: (3.7) H_{g}(m_{0}) = {m ∈ 𝒯 | ϱ(m, m_{0}) < ϱ(m, gm_{0})}. The remaining points in 𝒯 are denoted as: (3.8) E_{g}(m_{0}) = H_{g^{−1}}(gm_{0}). We shall refer to H_{g}(m_{0}) as a half space centered at m_{0} and to E_{g} as the exterior halfspace to m_{0} . (Of course, if we replace m_{0} by gm_{0} , then E_{g} is a halfspace centered at gm_{0} and H_{g} is exterior to gm_{0} .)

The Dirichlet polyhedron for Σ centered at m_{0} , D(m_{0}), is defined by: (3.9) D(m_{0}) = ∩{H_{g}(m_{0}) | g ∈ Σ^{∗}}. The closed Dirichlet polyhedron for Σ centered at m_{0} , Δ(m_{0}), is defined in a similar way:

(3.10) Δ(m_{0}) = ∩\{\overline{H}_{g}(m_{0}) | g ∈ Σ^{∗}\}.

The standard argument involving invariance of distance demonstrates that these sets are natural in the following sense:

(3.11) h(D(m_{0})) = D(h(m_{0})) for all h ∈ Σ, (3.12) h(Δ(m_{0})) = Δ(h(m_{0})) for all h ∈ Σ.

4. Equidistant loci and halfspaces We shall need to consider the closed halfspaces. The following fact is central

to our discussion.

Lemma 4.1. Suppose that m lies on L_{g}. Then the half-closed interval [m_{0}, m) is contained in H_{g}(m_{0}).

Proof. Suppose that L_{g}(m_{0}) ∩ [m_{0}, m) is nonempty. Let n be a point of this intersection. Then:

(1) ϱ(m_{0}, n) + ϱ(n, m) = ϱ(m_{0}, m),

(2) ϱ(m_{0}, n) = ϱ(g(m_{0}), n). On the other hand, since m is in L_{g}:

(3) ϱ(m_{0}, m) = ϱ(g(m_{0}), m). Thus, we conclude that:

(4) ϱ(g(m_{0}), n) + ϱ(n, m) = ϱ(g(m_{0}), m).

By Proposition 2.1, n lies on [m_{0}, g(m_{0})]. Clearly, n is not one of the endpoints of this interval. Hence, by uniqueness of geodesics, it follows that:

(5) γ(m_{0}, m) = γ(m_{0}, n) = γ(m_{0}, gm_{0}).

Now, γ(m_{0}, gm_{0}) is isometric to the real line with the standard metric. The points m and n are each equidistant from the distinct points of this line, m_{0} and gm_{0} . Clearly, this implies that m is equal to n. This is impossible, since n lies on [m_{0}, m).

Therefore, [m_{0}, m) is contained in the complement of L_{g}(m_{0}). The sets, H_{g}(m_{0}) and E_{g}(m_{0}), form a separation of this complement. Since [m_{0}, m) is connected and m_{0} is in H_{g}(m_{0}), [m_{0}, m) is contained in H_{g}(m_{0}).

Corollary 4.2. Suppose that m lies on L_{g}. Then the half-open interval from gm_{0} to m is contained in E_{g}(m_{0}).

Proof. This is immediate from (3.6), (3.8) and Lemma 4.1.

These results yield the desired description of the closed halfspaces.

Corollary 4.3.

(a) \overline{H}_{g}(m_{0}) = H_{g}(m_{0}) ∪ L_{g}(m_{0}) = \{m ∈ 𝒯 | ϱ(m, m_{0}) ≤ ϱ(m, gm_{0})\},

(b) \overline{E}_{g}(m_{0}) = E_{g}(m_{0}) ∪ L_{g}(m_{0}) = \{m ∈ 𝒯 | ϱ(m, gm_{0}) ≤ ϱ(m, m_{0})\},

(c) ∂H_{g} = ∂\overline{H}_{g} = L_{g} = ∂\overline{E}_{g} = ∂E_{g},

(d) H_{g} = int (\overline{H}_{g}) and E_{g} = int (\overline{E}_{g}).

Suppose that m is an element of a subset K of 𝒯 . We say that K is starlike at m if, for every point n in K, the geodesic segment from m to n is contained in K.

It is not clear whether the halfspaces are geodesically convex. This seems unlikely, except for the case of genus one and one puncture, where it is true, due to the fact that Teichmüller space with the Teichmüller metric is the hyperbolic plane. We shall be using the following weaker result:

Proposition 4.4. H_{g} and \overline{H}_{g} are starlike at m_{0} .

Proof. In view of Lemma 4.1 and Corollary 4.3(a), it suffices to show that H_{g} is starlike at m_{0} . To this end, let m be a point in H_{g}(m_{0}). Suppose [m_{0}, m] intersects L_{g}(m_{0}). Since L_{g} is closed and the interval is compact, we can choose n in [m_{0}, m] such that:

(1) n ∈ L_{g}(m_{0}) and (n, m] ⊂ H_{g}(m_{0}) ∪ E_{g}(m_{0}). By Lemma 4.1 and the connectedness of (n, m], we conclude that:

(2) [m_{0}, n) ∪ (n, m] ⊂ H_{g}(m_{0}). Observe that we have shown that, for every point p in H_{g}, the geodesic segment from p to m_{0} is contained in \overline{H}_{g}. Now consider the exponential map at m_{0} :

(3) Ω_{0}: Q(M_{0}) → 𝒯 . Let φ be the preimage of m, Ω_{0}(φ) = m. There exists a real number s such that 0 < s < 1 and Ω_{0}(sφ) = n.

Choose an open neighborhood 𝒰 of φ such that Ω_{0}(𝒰 ) is contained in H_{g}. The punctured cone on 𝒰 :

(4) 𝒞 (𝒰 ) = {tψ | 0 < t < 1, ψ ∈ 𝒰 } forms an open neighborhood of sφ in Q(M_{0}). By the observation above and Theorem 2.4, Ω_{0}(𝒞 (𝒰 )) is an open neighborhood of n which is contained in \overline{H}_{g}. Since n is in L_{g}, Corollary 4.3(c) implies that Ω_{0}(𝒞 (𝒰 )) meets E_{g}. This gives the desired contradiction.

This allows us to sharpen the assertion of Lemma 4.1:

Lemma 4.5. Suppose that m lies on L_{g}. Let r(m_{0}, m) be the geodesic ray from m_{0} through m. Then:

(a) r(m_{0}, m) ∩ H_{g}(m_{0}) = [m_{0}, m),

(b) r(m_{0}, m) ∩ L_{g}(m_{0}) = {m},

(c) r(m_{0}, m) ∩ E_{g}(m_{0}) = r(m_{0}, m) \ [m_{0}, m].

Proof. By Lemma 4.1 and the hypothesis:

(1) [m_{0}, m) ⊂ r(m_{0}, m) ∩ H_{g}(m_{0}),

(2) {m} ∈ r(m_{0}, m) ∩ L_{g}(m_{0}). In particular, each point in the closed interval [m_{0}, m] is contained in \overline{H}_{g}(m_{0}).

Suppose n is contained in a point on the ray which is not on this closed interval. Then m is in the half-closed interval [m_{0}, n). Hence, by Lemma 4.1, n is not in L_{g}. Likewise, by Proposition 4.4, n is not in H_{g}. Thus:

(3) r(m_{0}, m) \ [m_{0}, m] ⊂ r(m_{0}, m) ∩ E_{g}(m_{0}) = r(m_{0}, m). Since H_{g}, L_{g}, and E_{g} are disjoint, the result follows immediately.

Note: One does not need the full thrust of Theorem 2.4 for the proof of Proposition 4.4. An argument based on Corollary 4.3(c) and convergence of sequences of geodesics ([Bu, Chapter 1, (8.14)]) would lead to the same contradiction.

Let δ_{g} denote the difference ϱ_{gm_{0}} − ϱ_{m_{0}} of the distance functions, ϱ_{gm_{0}} and ϱ_{m_{0}} . Let J_{g} = γ(m_{0}, gm_{0}) \ [m_{0}, gm_{0}].

Lemma 4.6. δ_{g} is nonsingular exactly on the complement of J_{g} in 𝒯 .

Proof. By Corollary 2.7, δ_{g} is singular at m if and only if m_{0} and gm_{0} are on the same ray from m.

Proposition 4.7. L_{g} is a connected, properly embedded C^{1} -submanifold of codimension one in 𝒯 .

Proof. Since L_{g} is closed, it is properly imbedded.

Clearly, no point of L_{g} lies on J_{g}. Hence, by the implicit function theorem, L_{g} is a C^{1} -submanifold of codimension one in 𝒯 . In particular, L_{g} is bicollared in 𝒯 . (Alternatively, one can use the rays from m_{0} through L_{g} to define a global bicollar on L_{g}.)

Teichmüller space is contractible. By Proposition 4.4, Corollary 4.3 and (3.8), so are \overline{H}_{g} and \overline{E}_{g}. Since L_{g} is bicollared, a direct application of the Mayer– Vietoris sequence shows that L_{g} has the homology of a point. In particular, L_{g} is connected.

Proposition 4.8. If g and h are distinct elements of Σ, then L_{g} ∩ L_{h} is either empty or a properly embedded submanifold of codimension 2.

Proof. Let x be a point in L_{g} ∩ L_{h}. By the implicit function theorem, it suffices to show that the differentials at x of the functions δ_{g} and δ_{h} are linearly independent.

Let α be the unit norm quadratic differential at x pointing in the direction of m_{0} , and β_{1} (respectively β_{2} ) the unit norm quadratic differential at x pointing in the direction of gm_{0} (respectively hm_{0} ). Since the points m_{0} , gm_{0} and hm_{0} are distinct points in 𝒯 at equal distances from x, the rays from x to these points are distinct. Hence, the unit norm differentials, α, β_{1} , and β_{2} , are distinct.

By Corollary 2.7, the differential at x of δ_{g} (respectively δ_{h}) is equal to α−β_{1} (respectively α − β_{2} ). Hence, if these differentials were linearly dependent, α, β_{1} , and β_{2} would be affinely dependent. That is, they would lie on a common line (not necessarily passing through the origin). Since they are distinct unit vectors in Q(x), this would contradict the strict convexity of the unit sphere in Q(x).

We shall now prove some lemmas which will be used in establishing the polyhedral nature of D as well as in the proof that D is a locally finite fundamental domain.

Lemma 4.9. Let m be a point in 𝒯 and {g_{n}} be an infinite sequence of distinct mapping classes in Σ. Then the sequence of real numbers {ϱ(m, g_{n}m_{0})} is unbounded.

Proof. The argument is implicit in the proof of proper discontinuity, Theorem 2.11.

Lemma 4.10. For any compact set K in 𝒯 , the set {g ∈ Σ | K is not contained in H_{g}(m_{0})} is finite.

Proof. Suppose, for the sake of contradiction, that there exists an infinite sequence {g_{n}} of distinct mapping classes in Σ such that for every n we have K ∩ \overline{E}_{g_{n}}(m_{0}) ≠ ϕ. Choose a sequence of points, \{m_{n}\}, such that m_{n} is an element of K ∩ \overline{E}_{g_{n}}(m_{0}). Let m be an accumulation point of the sequence \{m_{n}\} in K. By taking a subsequence, we can assume that m_{n} converges to m. As m_{n} lies in \overline{E}_{g_{n}}(m_{0}), we have, for every n:

(1) ϱ(m_{n}, g_{n}m_{0}) ≤ ϱ(m_{n}, m_{0}).

Let ε be a positive real number. Since m_{n} converges to m, there exists an integer n_{0} such that, for every n ≥ n_{0} , the following two properties hold:

(2) ϱ(m_{n}, m_{0}) ≤ ϱ(m, m_{0}) + ε,

(3) ϱ(m, m_{n}) ≤ ε. By using the triangle inequality, we have:

(4) for all n ≥ n_{0} , ϱ(m, g_{n}m_{0}) ≤ ϱ(m, m_{0}) + 2ε, which shows that the sequence {ϱ(m, g_{n}m_{0})} is bounded. This contradicts Lemma 4.9.

The following corollary is an immediate consequence of the previous lemma.

Corollary 4.11. For any compact set K in 𝒯 , the set {g ∈ Σ | K∩L_{g}(m_{0}) ≠ ϕ} is finite.

5. The structure of the fundamental domain

The following few results give some information on the topology of D and Δ.

Proposition 5.1. m_{0} is in the interior of Δ.

Proof. Let K be a compact neighborhood of m_{0} in 𝒯 . By Lemma 4.10, there is a finite subset F of Σ such that:

(1) for each g ∈ Σ \ F, K ⊂ H_{g}(m_{0}).

For each element g in F, let K_{g} denote the intersection K ∩ H_{g}. Since H_{g} is open, K_{g} is a neighborhood of m_{0} in 𝒯 . The intersection (over all g in F) of the sets {K_{g}} is a neighborhood of m_{0} in 𝒯 which is contained in Δ.

Proposition 5.2. interior (Δ) = D and Δ = \overline{D}.

Proof. Let m be a point in the interior of Δ. In particular:

(1) for every g ∈ Σ^{∗}, ϱ(m, m_{0}) ≤ ϱ(m, gm_{0}). If, for some g in Σ^{∗}, equality held, then m would be on L_{g}(m_{0}). By Corollary 4.3(c), interior (Δ), being a neighborhood of m, would contain a point n satisfying the strict inequality, ϱ(n, m_{0}) > ϱ(n, gm_{0}), which is impossible since interior (Δ) is contained in Δ. Therefore, m is in D.

Conversely, let m be a point in D. By Lemma 4.10, we can find an open ball U containing m and a finite subset F of Σ such that, for each g in Σ \ F, U is contained in H_{g}(m_{0}). For each g in F, m is in H_{g}, which is open. Let U_{g} be equal to U ∩ H_{g}. The intersection of the sets {U_{g}} (over all g in F) is an open neighborhood of m contained in Δ. Hence, m is in the interior of Δ. This proves the first assertion.

For the second assertion, we know that \overline{D} is contained in Δ, because Δ is closed. Let us prove that Δ is contained in \overline{D}.

Let m be a point in Δ. By definition, for each g in Σ^{∗}, we have:

(2) m ∈ H_{g}(m_{0}) or m ∈ L_{g}(m_{0}).

Consider the geodesic segment [m_{0}, m] between m_{0} and m. By Lemma 4.1 and Proposition 4.4, (2) shows that, except possibly for its endpoint m, this segment is contained in D. Therefore, m is contained in \overline{D}.

Since intersections of sets which are starlike at m_{0} are starlike at m_{0} , we have an immediate corollary of Proposition 4.4.

Proposition 5.3. D(m_{0}) and Δ(m_{0}) are starlike at m_{0} .

With this, we may deduce the following analogue for ∂D of Proposition 4.7 (without the smoothness property). The proof follows the same plan as the proof of this previous proposition. (Note the remark in the proof concerning an alternative argument for the existence of a bicollar.)

Proposition 5.5. ∂D is a properly embedded, locally flat submanifold of codimension one in 𝒯 .

Let us recall that the property of being locally flat means the existence (locally) of a bicollar, that is, ∂D looks locally like R^{n} sitting in R^{n+1} . Of course, this topological definition allows for corners, as in the classical case.

The next two lemmas imply the existence of a fundamental set satisfying (3.2).

Lemma 5.6. For any point m in 𝒯 , the Σ-orbit of m intersects D(m_{0}) in at most one point.

Proof. Suppose that g_{1}(m) and g_{2}(m) are distinct points in D, where g_{1} and g_{2} are elements of Σ. Clearly, g_{1} and g_{2} are distinct. Hence, the assumption that g_{1} is in D(m_{0}) implies that:

(1) ϱ(g_{1}m, m_{0}) < ϱ(g_{1}m, g_{1}g_{2}^{−1}m_{0}) = ϱ(m, g_{2}^{−1}m_{0}) = ϱ(g_{2}m, m_{0}).

Similarly, since g_{2}m is in D:

(2) ϱ(g_{2}m, m_{0}) < ϱ(g_{1}m, m_{0}).

This gives the desired contradiction.

Lemma 5.7. For any element m of 𝒯 , there exists a mapping class h in Σ such that m is in Δ(hm_{0}).

Proof. Suppose, on the contrary, that m is not in Δ(hm_{0}) for any h in Σ. Then we can find an infinite sequence {g_{n}} of distinct mapping classes such that:

(1) ⋯ < ϱ(m, g_{2}m_{0}) < ϱ(m, g_{1}m_{0}) < ϱ(m, m_{0}). This implies that the sequence of real numbers {ϱ(m, g_{n}m_{0})} is bounded, in contradiction to Lemma 4.9.

The next result follows from Lemma 4.10.

Proposition 5.8. For any compact set K in 𝒯 , the set {g ∈ Σ | K ∩ Δ(gm_{0}) ≠ ∅} is finite.

Proof. If g is not the identity, then, by the definition of Δ(gm_{0}), (3.6), (3.8) and Corollary 4.3, we have:

(1) Δ(gm_{0}) ⊂ \overline{H}_{g^{−1}}(gm_{0}) = \overline{E}_{g}(m_{0}). Again, by Corollary 4.3, the result follows immediately from Lemma 4.10.

We have established all the necessary facts to deduce the main result.

Theorem 5.9. D is a locally finite fundamental domain for the action of Σ on 𝒯 .

Proof. Condition (3.1) is implicit in Propositions 5.2 and 5.3, (3.2) in Proposition 5.2 and Lemma 5.7, (3.3) in Proposition 5.5 and (3.4) in Proposition 5.8.

We shall close with a brief discussion of the polyhedral nature of D.

By Proposition 5.2, every point of ∂D lies on at least one hypersurface L_{g}(m_{0}). On the other hand, by Corollary 4.11, every point in 𝒯 lies on at most finitely many such hypersurfaces (for a fixed m_{0} ). We shall say that a point of ∂D lies on a face of D (or of Δ) if it lies on exactly one hypersurface. If g is an element of Σ, the g -face F_{g}(m) of D is the set of points of ∂D which lie only on the hypersurface L_{g}(m_{0}). (Therefore, m is on a face of D if and only if, for some g in Σ, m is on F_{g}.)

Suppose that x is a point in 𝒯 and n is in the Σ-orbit of m_{0} . We say that x is closest to n if ϱ(x, n) is less than or equal to ϱ(x, hm_{0}) for all h in Σ.

Proposition 5.10. Let x be a point in 𝒯 . Then x is on F_{g}(m_{0}) if and only if x is closest to exactly two points of the Σ-orbit of m_{0} , m_{0} and gm_{0} .

Proof. Given Proposition 5.2, the proof is a trivial exercise in the definitions.

Corollary 5.11. F_{g}(m_{0}) = F_{g^{−1}}(gm_{0}).

Corollary 5.12. F_{g}(m_{0}) ⊂ Δ(m_{0}) ∩ Δ(gm_{0}) ⊂ L_{g}(m_{0}).

Corollary 5.13. Let x be a point of Δ(m_{0}) ∩ Δ(gm_{0}). Then x is on a face of D(m_{0}) if and only if x is on a face of D(gm_{0}). If x is on a face of D(m_{0}), then x is on F_{g}(m_{0}).

We observe that, in particular, the property of being “on a face” is independent of which copy of D we consider. This, of course, is a property we would ask of any reasonable notion of a tesselation.

Proposition 5.14. There exists an element g of Σ such that F_{g}(m_{0}) is nonempty.

Proof. This follows immediately from Propositions 4.8 and 5.5. We shall, however, give more explicit information.

Let g be an element of Σ^{∗} such that m_{0} is closest to gm_{0} which, according to the definitions, means the following:

(1) ϱ(m_{0}, gm_{0}) ≤ ϱ(m_{0}, hm_{0}) for all h in Σ. Let m be the midpoint of [m_{0}, gm_{0}]. Suppose that m is not closest to m_{0} :

(2) ϱ(m, hm_{0}) < ϱ(m, m_{0}) for some h in Σ. Then, by the triangle inequality:

(3) ϱ(m_{0}, hm_{0}) ≤ ϱ(m_{0}, m) + ϱ(m, hm_{0}) < 2ϱ(m_{0}, m) = ϱ(m_{0}, gm_{0}). This contradicts the choice of g. Hence, m is closest to m_{0} :

(4) ϱ(m, m_{0}) = ϱ(m, gm_{0}) ≤ ϱ(m, hm_{0}) for all h in Σ. Thus, by Proposition 5.2, m is in ∂D ∩ L_{g}.

Suppose that m lies on another hypersurface L_{h}. Since m_{0} , gm_{0} and hm_{0} are distinct points equidistant from the midpoint of [m_{0}, gm_{0}], hm_{0} does not lie on γ(m_{0}, gm_{0}). Hence, by Proposition 2.1:

(5) ϱ(m_{0}, hm_{0}) < ϱ(m_{0}, m) + ϱ(m, hm_{0}) = ϱ(m_{0}, gm_{0}). Again, this contradicts the choice of g. Therefore, m lies on exactly one hypersurface L_{g}(m_{0}). This shows that m is in F_{g}(m_{0}).

Proposition 5.15. F_{g} is an open subset of ∂D.

Proof. Suppose that m is on F_{g}(m_{0}). By Lemma 4.10, we may choose a neighborhood U of m and a finite subset F of Σ such that U is contained in H_{h}(m_{0}) for all h in Σ \ F. For each element h of F \ {g}, let U_{h} denote the intersection U ∩H_{h}. Since m is in Δ and lies on only one hypersurface L_{g}, U_{h} is an open neighborhood of m for each such h. The intersection V of the sets {U_{h}} (over h in F \ {g}) is an open neighborhood of m which is contained in H_{h}(m_{0}) for all h in Σ \ {g}. The intersection of V with ∂D gives a neighborhood of m in ∂D entirely contained in F_{g}.

Proposition 5.16. The union of the faces of ∂D is a dense open subset of ∂D.

Proof. The complement of this union consists of points which, by Proposition 4.8 and Corollary 4.11, lie on a locally finite union of codimension one submanifolds of ∂D. This set is closed and nowhere dense. Hence, the union is a dense open subset.

We note that the structure which we have developed is sufficient to deduce that the group Σ is generated by face pairing transformations. We say that an element g of Σ is a face pairing transformation of D(m_{0}) if F_{g}(m_{0}) is nonempty. By Proposition 4.8, Corollary 5.12 and Proposition 5.15, this amounts to the assertion that Δ(m_{0}) ∩ Δ(gm_{0}) has nonempty interior in ∂D(m_{0}).

Theorem 5.17. Σ is generated by the face pairing transformations of D(m_{0}).

Proof. The proof is analogous to one that can be done in the case of discrete isometry groups acting in hyperbolic spaces:

Let h be an element of Σ. Clearly, m_{0} is in D(m_{0}) and hm_{0} is in D(hm_{0}). Let α be a path in 𝒯 from m_{0} to hm_{0} . By Propositions 5.1, 5.2, 5.5 and 5.8, we may assume that, for all g in Σ, α is transverse to ∂D(gm_{0}) and intersects ∂D(gm_{0}) only in points on faces of D(gm_{0}).

As we traverse α, we encounter various copies of our domain:

(1) D(h_{i}m_{0}), 0 ≤ i ≤ n, such that:

(2) h_{0} = 1 and h_{n} = h,

(3) ∅ ≠ F_{g_{i}}(h_{i}m_{0}) ⊂ Δ(h_{i}m_{0}) ∩ Δ(h_{i+1}m_{0}), 1 ≤ i ≤ n, where:

(4) h_{i+1} = g_{i}h_{i}. We prove, by induction, that h_{i} is a product of face pairing transformations of D(m_{0}) for 1 ≤ i ≤ n.

By definition, h_{1} is a face pairing transformation. Suppose that h_{i} is a product of face pairing transformations. By conjugating (3), we obtain:

(5) ∅ ≠ F_{h_{i}^{−1}h_{i+1}}(m_{0}) ⊂ Δ(m_{0}) ∩ Δ(h_{i}^{−1}h_{i+1}m_{0}).

This implies that h_{i}^{−1}h_{i+1} is a face pairing transformation. But:

(6) h_{i+1} = h_{i}(h_{i}^{−1}h_{i+1}). Hence, h_{i+1} is a product of the desired type.

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Received 19 September 1994