ON THE ACTION OF THE MAPPING CLASS GROUP FOR RIEMANN SURFACES OF INFINITE TYPE

EGE FUJIKAWA, HIROSHIGE SHIGA AND MASAHIKO TANIGUCHI

Abstract. We consider Riemann surfaces of infinite type and their reduced Teichmüller spaces. The reduced Teichmüller space admits the action of the reduced mapping class group. Generally, the action is not discrete while it is faithful. We give sufficient conditions for the discreteness of the action in terms of the geometry of Riemann surfaces.

1. Introduction

The mapping class group (or modular group) Mod(R) for a Riemann surface R is the set of equivalence classes of quasiconformal self-maps of R (cf. [7]). Two quasiconformal self-maps h_{1} and h_{2} of R are equivalent if h_{2}^{−1} ∘ h_{1} is homotopic to the identity by a homotopy that keeps every points of ideal boundary ∂R fixed throughout. In the theory of Teichmüller spaces of Riemann surfaces of analytically finite type, the mapping class group plays an important role in various fields. This is a group of the biholomorphic automorphisms of the Teichmüller space and it acts faithfully and discontinuously. On the other hand, it seems that there are few studies on Mod(R) for a Riemann surface R of infinite type. Recently, Earle-Gardiner-Lakic showed in [3] that it acts faithfully on T (R). In this paper, we consider the discreteness of the action of the mapping class group. We say that a subgroup G of Mod(R) is discrete if the orbit of any point of T (R) under the G-action is discrete.

For a Riemann surface of analytically finite type, Mod(R) is discrete, while in the case of infinite type, Mod(R) is not necessarily discrete. In particular, if R has a boundary curve (border), Mod(R) is not discrete since a slight change of the boundary value of a quasiconformal map produces a different mapping class in Mod(R). Thus, it is natural that we consider another group, the reduced mapping class group. The reduced mapping class group Mod^{#}(R) is the set of homotopy classes of quasiconformal self-maps of R whose homotopy maps does not necessarily keep points of ∂R fixed. The reduced mapping class group is also important because it naturally acts on the reduced Teichmüller space.

We explore the problem of discreteness of the reduced mapping class group for Riemann surfaces of infinite type. Actually, if R is a Riemann surface of topologically finite type, then Mod^{#}(R) is discrete. However, Mod^{#}(R) is not discrete in general. For example, if R has a sequence of disjoint simple closed geodesics which are not freely homotopic to a boundary component and whose lengths tend to 0, then we see that Mod^{#}(R) is not discrete (See §3 and §6). The purpose of this paper is to give a sufficient condition for discreteness.

The authors thank Katsuhiko Matsuzaki for his valuable suggestions.

Date: November 9, 2001.

2. The mapping class group for the reduced Teichmüller space

Throughout this paper, we assume that a Riemann surface R is hyperbolic, that is, it is represented by ℍ/Γ for some Fuchsian group Γ acting on the upper half plane ℍ. We also assume that the Fuchsian group Γ is always non-elementary. In other words, we assume that the group Γ is non-abelian. A Riemann surface is called of analytically finite type if the hyperbolic area is finite, and is called of analytically infinite type if the area is not finite.

For an open Riemann surface R, a relatively non-compact connected component of the complement of a compact subset of R is called an end. An end V of R is called a hole if it is doubly connected and the hyperbolic area of V is infinite. A doubly connected end of R is called a cusp if the hyperbolic ares of V is finite. A cusp V with smooth relative boundary is conformally equivalent to the punctured disk {0 < |z| < 1}. An ideal boundary of R corresponding to the origin z = 0 is called a puncture.

Notation. The hyperbolic distance between A, B ⊂ ℍ is denoted by d_{ℍ}(A, B) and the hyperbolic length of a curve c in ℍ or a Riemann surface R is denoted by ℓ(c).

We review the theory of Teichmüller spaces and the mapping class group. See [4], [5] and [7] for the details.

Definition 1. Fix a Riemann surface R. For a pair (S, f) of a Riemann surface S and a quasiconformal map f of R onto S, we say that (S_{1}, f_{1}) and (S_{2}, f_{2}) are RT (reduced Teichmüller) equivalent if there exists a conformal map h of S_{1} onto S_{2} such that f_{2}^{−1} ∘ h ∘ f_{1} is homotopic to the identity on R.

The reduced Teichmüller space T^{#}(R) with the base Riemann surface R is the set of all RT equivalence classes [S, f] of such pairs (S, f) as above.

Definition 2. We say that two quasiconformal self-maps h_{1} and h_{2} of R are RT equivalent if h_{2}^{−1} ∘ h_{1} is homotopic to the identity on R.

The reduced mapping class group Mod^{#}(R) is the set of all RT equivalence classes [h] of quasiconformal self-maps h of R.

Every quasiconformal map of R = ℍ/Γ induces an isomorphism of Γ into PSL(2, ℝ). We see that if two self-maps h_{1} and h_{2} are RT equivalent then they induce the same isomorphism modulo PSL(2, ℝ) conjugacy.

If R is a compact Riemann surface, then the reduced Teichmüller space T^{#}(R) is nothing but the ordinary Teichmüller space T (R) of R and the reduced mapping class group Mod^{#}(R) is the ordinary mapping class group Mod(R).

Similar to the case of T (R), the reduced Teichmüller space T^{#}(R) is equipped with the reduced Teichmüller distance d_{T} (·, ·) defined by

d_{T} ([S, f], [S′, g]) = \frac{1}{2} inf_{f, g} log K(f ∘ g^{−1}),

where K(·) is the maximal dilatation of a quasiconformal map and the infimum is taken over all quasiconformal maps f and g determining [S, f] and [S′, g], respectively. It is known that T^{#}(R) is a complete metric space with respect to this d_{T}. An element ω = [h] ∈ Mod^{#}(R) induces an automorphism of T^{#}(R) by

[S, f] ↦ [S, f ∘ h^{−1}].

This is an isometric automorphism with respect to d_{T} and denoted by ω_{∗}. Namely, we have a homomorphism Mod^{#}(R) → Aut T^{#}(R).

Remark 1. In [3], it is proved that for any Riemann surface R of infinite analytic type (and if 2g + n > 4 when R is of finite (g, n)-type), the homomorphism Mod^{#}(R) → Aut T^{#}(R) as above is faithful. Therefore we can identify ω_{∗} with ω and omit the asterisk hereafter.

Definition 3. We say that a subgroup G of Mod^{#}(R) is discrete if every sequence {ω_{n}} ⊂ G satisfying lim_{n→∞} ω_{n}(p) = q for some pair of points p, q in T^{#}(R) is eventually a constant sequence, that is, there exists an N ∈ ℕ such that ω_{n} = ω_{N} for every n ≥ N.

3. Examples

As we noted in the introduction, if R is a compact Riemann surface, then the action of Mod(R) on T (R) is discrete. Contrary to this case, there are various kinds of examples which show non-discreteness of Mod^{#}(R) for a Riemann surface R of infinite type.

Example 1. Suppose that R has a sequence {c_{n}} of disjoint simple closed geodesics that are not peripheral (i.e. that are not freely homotopic to a boundary component) and that these hyperbolic lengths tend to 0. Then the Dehn twist along each c_{n} gives an element ω_{n} of Mod^{#}(R) such that the sequence {ω_{n}(p_{0})} converges to p_{0} as n → ∞, where p_{0} = [R, id] is the base point of T^{#}(R). Hence Mod^{#}(R) is not discrete.

There exists a Riemann surface R without short geodesics but containing a point with arbitrarily large injectivity radius with respect to the hyperbolic metric such that Mod^{#}(R) is not discrete.

Example 2. Set

R = ℂ − ⋃_{n=1}^{∞} ⋃_{m∈ℤ} \{\frac{m}{n} + (2n + 1)\sqrt{−1}\} .

Then we can see that R has no short geodesics by considering the classical extremal domains, whereas the injectivity radii at 2ni tend to ∞.

Further, set

f_{n}(z) = \{ \begin{matrix}x − (y − 2n − 2)/n + y\sqrt{−1} & (2n + 1 ≤ y < 2n + 2) \\ x + (y − 2n)/n + y\sqrt{−1} & (2n ≤ y < 2n + 1) \\ x + y\sqrt{−1} & elsewhere. \end{matrix}

Then f_{n} are quasiconformal self-maps of R and the maximal dilatations of {f_{n}} tend to 1. Thus Mod^{#}(R) is not discrete.

There is an another example of a planar Riemann surface R without cusps but containing a point with arbitrarily large injectivity radius with respect to the hyperbolic metric such that Mod^{#}(R) is not discrete.

Example 3. For each n ≥ 2, we set

I_{n} = [−1, 1] ∪ ⋃_{k=1}^{∞} I_{n,k},

where

I_{n,k} = \{x + (1 − 1/n)^{k}\sqrt{−1} | −1 ≤ x ≤ 1\}
⋃ \{x + (1 + (k − 1)/n)\sqrt{−1} | −1 ≤ x ≤ 1\}.

We take infinitely many copies \{R_{n}\} of ℂ−\{y\sqrt{−1} | y ≤ −1\} and set R_{n}′ = R_{n}−I_{n} for each n ≥ 2. We make a Riemann surface R by gluing the right hand side of \{y\sqrt{−1} | y < −1\} on R_{n}′ with the left hand side of \{y\sqrt{−1} | y < −1\} on R_{n−1}′ (n = 3, 4, …) along the imaginary axis.

Consider a quasiconformal map f_{n} of R_{n}′ defined by

f_{n}(z) = \{ \begin{matrix}x + (1 − 1/n)y\sqrt{−1} & (0 < y ≤ 1) \\ x + (y − 1/n)\sqrt{−1} & (y > 1) \\ x + y\sqrt{−1} & elsewhere. \end{matrix}

It is easily seen that the maximal dilatations of f_{n} converge to 1 as n → ∞. Obviously, f_{n} is extended to a quasiconformal self-map of R by setting it the identity on R−R_{n}′ and we denote it by the same letter f_{n}. Thus the quasiconformal map f_{n} gives an element [f_{n}] of Mod^{#}(R) such that {[f_{n}](p_{0})} converges to p_{0} as n → ∞, where p_{0} = [R, id] is the base point of T^{#}(R). Hence we conclude that Mod^{#}(R) is not discrete.

Even if a Riemann surface R has no short geodesics and no points with arbitrarily large injectivity radius, Mod^{#}(R) may not be discrete as the following example shows:

Example 4. Consider a torus S with two geodesic borders with the same length to each other. We take infinity many copies {S_{n}}_{n=−∞}^{∞} of S. We denote the two geodesic borders of S_{n} by ℓ_{n,1} and ℓ_{n,2}. Construct a Riemann surface R by gluing the ℓ_{n−1,2} with ℓ_{n,1} and gluing ℓ_{n,2} with ℓ_{n+1,1} for each n. Let f be a conformal self-map of R which sends a point a ∈ S_{n} to a point of S_{n+1} corresponding to the same point of S as a, and we set f_{n} := f^{n}. Then we see that [f_{n}] ≠ id as an element of Mod^{#}(R). However, [f_{n}](p_{0}) = p_{0} for all n, where p_{0} = [R, id] ∈ T^{#}(R) because f_{n} : R → R is a conformal mapping. Hence, Mod^{#}(R) is not discrete.

4. Main Results

As Example 1 shows, for the discreteness of the mapping class group, it is necessary that there exists no sequence of geodesics on the Riemann surface whose lengths converge to zero. Example 2, 3 show that some conditions for the injectivity radius are required for the discreteness.

Definition 4. For a given M > 0, we say that a point p of R belongs to a subset R_{M} of R if there exists a non-trivial simple closed curve c_{p} containing p such that the hyperbolic length of c_{p} is less than M. The set R_{ϵ} is called the ϵ-thin part of R if ϵ > 0 is smaller than the Margulis constant.

Now, we exhibit our main results.

Theorem 1. Let R be a Riemann surface with non-abelian fundamental group. Suppose that R satisfies the following two conditions

1. There exists an ϵ > 0 such that the ϵ-thin part of R consists only of cusp neighborhoods. (i.e. the ϵ-thin part equals to the ϵ-cuspidal part.)

2. There exist a constant M and a connected component R_{M}^{∗} of R_{M} such that the natural map of π_{1}(R_{M}^{∗}) to π_{1}(R) which is induced by the inclusion map of R_{M}^{∗} to R is surjective.

For a simple closed geodesic c on R, we set

Mod_{c}^{#}(R) = {[f] ∈ Mod^{#}(R) | f(c) is freely homotopic to c}.

Then Mod_{c}^{#}(R) is discrete on T^{#}(R).

Remark 2. Example 1 shows that the first condition of Theorem 1 is necessary for the discreteness, while Example 2 shows that it is not sufficient for the discreteness of Mod_{c}^{#}(R). The Riemann surface R in Example 4 satisfies both conditions of Theorem 1 but Mod^{#}(R) is not discrete.

Remark 3. The region R_{M} is not necessarily connected for large M even if the natural map : π_{1}(R_{M}) → π_{1}(R) is surjective. Moreover, in Example 7 of §6, we give a Riemann surface R and divergent sequences {M_{n}}, {M′_{n}} such that

• M_{n} < M′_{n} < M_{n+1} < M′_{n+1} (n = 1, 2, …).

• The natural map : π_{1}(R_{M_{n}}) → π_{1}(R) is surjective and R_{M_{n}} is connected for all n.

• The natural map : π_{1}(R_{M′_{n}}) → π_{1}(R) is surjective but R_{M′_{n}} is not connected for all n.

For a hyperbolic Riemann surface R = ℍ/Γ, we consider the convex core C(Γ) of the limit set of Γ, that is, the hyperbolic convex envelope of Λ(Γ) ⊂ ℝ ∪ {∞} in ℍ. Since the convex core C(Γ) is Γ-invariant, it determines a region C(R) in R and we call the region the convex core of R.

Definition 5. We say that a Riemann surface R has ϵ-uniform geometry if the following two conditions are satisfied for some ϵ > 0:

1. The ϵ-thin part of R consists of cusp neighborhoods.

2. The injectivity radius on the convex core C(R) of R is less than ϵ^{−1}.

Since C(R) is connected and it contains any closed geodesic on R, from Theorem 1 we have the following immediately .

Corollary 1. Let R be a Riemann surface with ϵ-uniform geometry for some ϵ > 0. Then Mod_{c}^{#}(R) is discrete for any simple closed geodesic c on R.

Remark 4. The conditions in Theorem 1 do not imply the uniform geometry. For example, set R = ℂ − {n : n ∈ ℤ}. Then R has a Fuchsian model of the first kind, and hence the convex core C(R) coincides with R. But since R has points with arbitrarily large injectivity radius, R does not have ϵ-uniform geometry for any ϵ > 0. On the other hand, it is easily seen that R satisfies the conditions in Theorem 1.

Remark 5. The conditions having uniform geometry were first stated as no short geodesics and no large disk condition. Nakanishi and Yamamoto[8] shows that under these conditions the out radius of the Teichmüller space is strictly less than 6. Ohtake[9] uses these conditions to show that the norm of the Poincaré series is strictly less than one which generalizes a result in McMullen[6].

It is important to give conditions for the mapping class group to be discrete. By using the above results, we have the following.

Theorem 2. Let R be a Riemann surface satisfying the conditions in Theorem 1 or Corollary 1. Suppose that either the genus of R, the number of cusps or the number of holes is positive finite. Then Mod^{#}(R) is discrete.

For a planar Riemann surface, Theorem 2 does not necessarily hold. See Example 6 in §6.

5. Proofs of main results

First of all, we note the geometry of a component of R_{M}.

Proposition 1. For M > 0, let R_{M}^{∗} be a connected component of R_{M} defined in Definition 4 and R_{ϵ} the ϵ-thin part of R for some small ϵ < M. We assume that R_{M}^{∗} − R_{ϵ} is not of type (0, 3). Then there exists a constant M_{1} > 0 depending only on M and ϵ such that for any point p ∈ R_{M}^{∗} − R_{ϵ} there exists a simple closed curve c_{p} passing through p with ℓ(c_{p}) < M_{1} which does not surround a puncture of R.

Proof. Let Γ be a Fuchsian group representing R. Take an arbitrary point p in R_{M}^{∗} − R_{ϵ}. From the definition, we may find a simple closed curve c_{p} ∋ p whose length is less than M. If c_{p} is not homotopic to a simple closed curve which surrounds a puncture of R, then there is nothing to prove.

Thus, we suppose that c_{p} surrounds a puncture of R. Then, a parabolic transformation γ ∈ Γ represents c_{p}. We may assume that γ(z) = z + 1. We take δ(ϵ), δ(M) > 0 so that

d_{ℍ}(δ(ϵ)\sqrt{−1}, δ(ϵ)\sqrt{−1} + 1) = ϵ,
d_{ℍ}(δ(M)\sqrt{−1}, δ(M)\sqrt{−1} + 1) = M.

We put

S(M, ϵ) = {z ∈ ℍ | δ(M) ≤ Im z ≤ δ(ϵ), 0 ≤ Re z ≤ 1}.

Since ℓ(c_{p}) < M and p ∉ R_{ϵ}, a lift C_{p} of c_{p} contains a point in S(M, ϵ).

Let L_{z} (z ∈ ℍ) denote the geodesic arc from z to z + 1. If the projection l_{z} in R of L_{z} via the canonical projection π : ℍ → R = ℍ/Γ is not simple for some z ∈ S(M, ϵ), then l_{z} contains a non-trivial simple closed curve c′_{z} with ℓ(c′_{z}) < ℓ(l_{z}) < M. Noting that l_{z} is the projection of the geodesic arc L_{z}, we verify that c′_{z} is not homotopic to c_{p}. Connecting c′_{z} and c_{p}, we have a simple closed curve passing through p with length less than M_{1} = 2(M + d_{ℍ}(δ(ϵ)\sqrt{−1}, δ(M)\sqrt{−1})) which does not surround a puncture of R.

Finally, we suppose that l_{z} is simple for any z ∈ S(M, ϵ). Let us consider a geodesic L_{z} for z ∈ R̃_{M}^{∗} ∩ {z ∈ ℍ | Im z = δ(M)}, where R̃_{M}^{∗} is a lift of R_{M}^{∗} with R̃_{M}^{∗} ∩ S(M, ϵ) ≠ ∅. From the definition, ℓ(L_{z}) = M. Therefore, there exists a simple closed curve c_{z} in R_{M}^{∗} passing through π(z) with ℓ(c_{z}) < M = ℓ(L_{z}) = ℓ(l_{z}). Obviously, the curve c_{z} is not homotopic to l_{z} = π(L_{z}) because l_{z} is the shortest simple closed curve which passes through π(z) and surrounds the puncture. Therefore, connecting c_{z} and c_{p} as before, we have a non-trivial simple closed curve passing through p with length less than M_{1} not surrounding a puncture of R.

To prove the main results, the following proposition on hyperbolic geometry is crucial.

Proposition 2. Let Γ be a Fuchsian model on the upper half plane ℍ of a Riemann surface R. Assume that Γ is non-elementary.

Let M and D be positive constants, and g_{1}, g_{2} and g_{3} be distinct hyperbolic elements in Γ such that

1. translation lengths of g_{j} (j = 1, 2, 3) are less than M,

2. the projections of the axes ℓ_{j} of g_{j} to R are simple closed geodesics, and 3. the distances between a point z_{1} on ℓ_{1} and ℓ_{j} (j = 2, 3) are less than D. Let f be a quasiconformal self-map of ℍ such that f ∘ Γ ∘ f^{−1} = Γ, and χ an

isomorphism of Γ induced by f.

Suppose that

χ(g_{1}) = g_{1}, χ(g_{2}) = g_{2}, χ(g_{3}) ≠ g_{3}.

Then there is a constant A > 1 depending only on M and D such that

K(f) ≥ A.

To prove this proposition, we prepare some well-known results (cf. [1], [5]).

Lemma 1 (Teichmüller). Let f be a quasiconformal selfmap of ℂ fixing 0 and 1, and suppose that there is a point z_{0} in ℂ − {0, 1} such that

log M = d_{1}(z_{0}, f(z_{0})) > 0.

Then K(f) ≥ M^{2}, where d_{1}( , ) is the hyperbolic distance on ℂ − {0, 1}.

Lemma 2 (Wolpert). Let f be a quasiconformal mapping of a Riemann surface R onto an another Riemann surface S, and c be a simple closed geodesic on R with hyperbolic length L. Then the hyperbolic length of a closed geodesic on S homotopic to f(c) is not greater than K(f)L.

Lemma 3 (Collar lemma 1). For a given M > 0, let g and g′ be arbitrary two hyperbolic elements of Γ with translation lengths less than M. Suppose that the projections of the axes of g and g′ to R are simple closed geodesics which intersect to each other. Then the axes make an angle greater than C > 0 depending only on M.

Lemma 4 (Collar lemma 2). For a given M > 0, let g and g′ be arbitrary two distinct hyperbolic elements of Γ with translation lengths less than M. Suppose that the projections of the axes of g and g′ to R are simple closed geodesics which coincide or disjoint. Then the axes have a distance greater than C > 0 depending only on M.

Proof of Proposition 2. We may assume that fixed points of g_{1} are 0 and ∞, and that z_{1} = \sqrt{−1} ∈ ℍ, hence d_{ℍ}(\sqrt{−1}, ℓ_{j}) ≤ D for j = 2, 3. We may also assume that the maximal dilatation of f is less than 2. Then at least one of the fixed points of g_{j} (j = 2, 3) is not in U = {x ∈ ℝ | |x| < δ or |x| > 1/δ} for sufficiently small δ > 0 which depends only on M and D. Indeed, if both fixed points of g_{j} are in U_{1} = \{x ∈ ℝ | |x| < δ\} for small δ > 0, then it contradicts d_{ℍ}(\sqrt{−1}, ℓ_{j}) ≤ D. The same argument works when both fixed points are in U_{2} = {x ∈ ℝ | |x| > 1/δ}. Next, suppose that one fixed point of g_{j} is in U_{1} and the other is in U_{2}. If ℓ_{1}∩ℓ_{j} ≠ ∅, then it contradicts Lemma 3. If ℓ_{1} ∩ ℓ_{j} = ∅, it contradicts Lemma 4. Therefore, we verify that at least one of the fixed points of g_{j} (j = 2, 3) is not in U. By using the same argument, we see that there exists a constant δ′ > 0 depending only on M and D such that all fixed points of g_{2} and g_{3} are in {x ∈ ℝ | δ′ < |x| < 1/δ′}.

From the above consideration, we verify that the Euclidean diameter diam(ℓ_{3}) of ℓ_{3} is greater than some r = r(M, D) > 0 which depends only M and D. Set g_{4} = f ∘ g_{3} ∘ f^{−1}. By the assumption we have g_{4} ≠ g_{3}. Then, we see that there exists a constant C = C(M, D) > 0 depending only on M and D such that an inequality

|b − f(b)| > C (1)

holds for a fixed point b of g_{3}. Indeed, since K(f) < 2, the translation length of g_{4} is less than 2M by Lemma 2. Noting that diam(ℓ_{3}) > r, we see that if ℓ_{3} ∩ ℓ_{4} ≠ ∅, then we have the assertion from Lemma 3 and that Lemma 4 yields the assertion if ℓ_{3} ∩ ℓ_{4} = ∅.

Take a fixed point a of g_{2} with

δ < |a| < \frac{1}{δ}. (2)

Let ϕ be a Möbius transformation with ϕ(0) = 0, ϕ(a) = 1 and ϕ(∞) = ∞. As we noted,

δ′ < |b| < \frac{1}{δ′} .

Hence, (1) and (2) implies that

d_{a}(b, f(b)) > log L

holds for some constant L > 1 depending only on M and D, where d_{a}(,) is the hyperbolic distance on ℂ − {0, a}. Considering {0, a, b} instead of {0, 1, z_{0}} for z_{0} = ϕ(b) in Lemma 1, we verify that the assertion follows for A = L^{2}.

Next, we show a fundamental property of Mod_{c}^{#}(R).

Proposition 3. Let R be a Riemann surface. For an arbitrary simple closed geodesic c, let {[f_{n}]} be a sequence of transformations of Mod_{c}^{#}(R) that satisfies lim_{n→∞} K(f_{n}) = 1. Then there exists a subsequence {[f_{n_{j}}]} of {[f_{n}]} such that {f_{n_{j}}} locally uniformly converges to a conformal self-map f of R which determines an transformation [f] ∈ Mod_{c}^{#}(R).

Proof. Let Γ be a Fuchsian model of R. First we suppose that c is not homotopic to a boundary component of R. Then there exists a simple closed geodesic c′ on R with c ∩ c′ ≠ ∅. Hence Lemma 5 (with K = c and C = c′) below shows the desired result.

Next suppose that c is homotopic to a boundary component of R. We may assume that the Riemann surface R is not topologically finite. Hence, there exists a pair of pants S in R such that the boundary consists of dividing simple closed geodesics c_{1} and c_{2} other than c, and neither c_{1} nor c_{2} is homotopic to a boundary component of R. If f_{n}(c_{1}) are homotopic to c_{1} or c_{2} for infinitely many n, then we can apply the previous argument. Hence we assume that f_{n}(c_{1}) are homotopic to neither c_{1} nor c_{2} for all n. Since f_{n}(c) are homotopic to c, and f_{n}(S) are still pairs of pants bounded by f_{n}(c), f_{n}(c_{1}) and f_{n}(c_{2}) for all n, we conclude that f_{n}(c_{1})∩S̄ ≠ ∅ or f_{n}(c_{2}) ∩ S̄ ≠ ∅. We may assume that f_{n}(c_{1}) ∩ S̄ ≠ ∅. Then the following Lemma 5 again shows the desired result.

Lemma 5. Let {f_{n}} be a sequence of quasiconformal self-maps of a hyperbolic Riemann surface R that satisfies lim_{n→∞} K(f_{n}) = 1. Suppose that there exist compact subsets C and K of R such that f_{n}(C) ∩ K ≠ ∅ for all n. Then there exist a subsequence {f_{n_{j}}} of {f_{n}} and a conformal self-map f of R such that {f_{n_{j}}} converge to f locally uniformly on R.

Proof. From the assumption, there exists a sequence {p_{n}} on C such that f_{n}(p_{n}) ∈ K. Since C and K are compact, there exist p ∈ C and q ∈ K such that p_{n} → p and f_{n}(p_{n}) → q as n → ∞ Take lifts of p_{n}, p and q in ℍ, say p̃_{n} p̃ and q̃, respectively, so that p̃_{n} → p̃ as n → ∞. We can take lifts f̃_{n} : ℍ → ℍ of f_{n} satisfying f̃_{n}(p̃_{n}) → q̃. Since {f̃_{n}} is a normal family, a subsequence {f̃_{n_{j}}} of f̃_{n} converges locally uniformly, and the limit function f̃ is either a quasiconformal self-map of ℍ or a constant in ℝ ∪ {∞}. Since f̃(p̃) = q̃ is in ℍ, f̃ is not a constant. Thus, it follows from lim_{n→∞} K(f_{n}) = 1 that f̃ is a conformal self-map of ℍ. Hence, {f_{n_{j}}} converges locally uniformly to a conformal self-map f of R which is the projection of f̃.

Before proving our main theorems, we shall give a sufficient condition for discreteness of a sequence of Mod^{#}(R) under the conditions in Theorem 1.

Proposition 4. Let R be a Riemann surface satisfying the two conditions in Theorem 1, and {f_{n}} be a sequence of quasiconformal self-maps of R satisfying the following conditions:

• {(f_{n})_{∗}} converges to the identity, where (f_{n})_{∗} : π_{1}(R) → π_{1}(R) is an isomorphism induced by f_{n}.

• lim_{n→∞} K(f_{n}) = 1. Then, as an element of Mod^{#}(R), [f_{n}] = [id] for sufficiently large n.

Proof. Let Γ be a Fuchsian model of R, and f̃_{n} a lift of f_{n} for each n. We may take f̃_{n} so that the isomorphisms χ_{n} : Γ → Γ induced by f̃_{n} converge to the identity.

Since the natural map π_{1}(R_{M}^{∗}) → π_{1}(R) is surjective for sufficiently large M > 0, we may find two distinct simple closed geodesics L_{1}^{0} and L_{2}^{0} on R_{M}^{∗} so that their lengths are less than M. Let γ_{j} (j = 1, 2) be hyperbolic elements of Γ which represent L_{j}^{0}. Since χ_{n} → id (n → ∞), χ_{n}(γ_{1}) = γ_{1} and χ_{n}(γ_{2}) = γ_{2} for sufficiently large n.

Suppose that χ_{n} are not eventually the identity. Then we prove the proposition by drawing a contradiction. We may find a γ_{n} ∈ Γ so that χ_{n}(γ_{n}) ≠ γ_{n}. The following lemma shows more, that is, we may take better one as γ_{n}.

Lemma 6. For sufficiently large n, there exists a hyperbolic element γ_{n} of Γ that satisfies the following two conditions:

1. χ_{n}(γ_{n}) ≠ γ_{n}.

2. the projection of the axis of γ_{n} on R is a simple closed geodesic with length less than M.

Proof. Since χ_{n} ≠ id, there exists an element α_{n} of Γ such that χ_{n}(α_{n}) ≠ α_{n}. We will show that either α_{n} ∘γ_{1} ∘α_{n}^{−1} or α_{n} ∘γ_{2} ∘α_{n}^{−1} is a desired element. It is obvious that both of them satisfy the second condition of the lemma. Hence, it suffices to show that one of them satisfies the first condition.

Suppose that χ_{n} fixes α_{n} ∘ γ_{j} ∘ α_{n}^{−1} (j = 1, 2). Then

β_{n} ∘ γ_{j} ∘ β_{n}^{−1} = γ_{j}, (j = 1, 2)

where β_{n} = α_{n}^{−1} ∘ χ_{n}(α_{n}). Thus, β_{n} fixes all fixed points of γ_{1} and γ_{2}. Since γ_{1} and γ_{2} are non-commutative, the Möbius transformation β_{n} fixes four points and it must be the identity map. This contradicts χ_{n}(α_{n}) ≠ α_{n}.

Now, under the assumption of Theorem 1, we show the existence of elements in Γ satisfying the assumption of Proposition 2 for χ_{n}.

Lemma 7. For each n, there exist hyperbolic elements g_{j,n} (j = 1, 2, 3) with axes ℓ_{j,n} such that they satisfy the following four conditions.

1. the projections L_{j,n} of ℓ_{j,n} to R are simple closed geodesics.

2. there is a constant M independent of n such that the lengths of L_{j,n} are less than M.

3. there is a constant D independent of n such that the distances between a point on ℓ_{1,n} and ℓ_{j,n} (j = 2, 3) are less than D, and

4. χ_{n}(g_{j,n}) = g_{j,n} for j = 1, 2, and χ_{n}(g_{3,n}) ≠ g_{3,n}.

Proof. Let γ_{n} be an element in Lemma 6. We denote by ℓ_{1}^{0}, ℓ_{2}^{0} and ℓ_{n} the axes of γ_{1}, γ_{2} and γ_{n}, respectively. Let R̃_{M}^{∗} be a lift of R_{M}^{∗}.

Fix a point z_{1} on ℓ_{1}^{0}. There exists the nearest point z_{n} on ℓ_{n} from z_{1}. Since z_{1} and z_{n} belong to R̃_{M}^{∗}, and R̃_{M}^{∗} is connected, there exists a curve C_{n} on R̃_{M}^{∗} which connects z_{1} and z_{n}. Furthermore, we can take the curve C_{n} so that the projection of C_{n} is in R_{M}^{∗} − R_{ϵ}.

First, we observe a fundamental property of R_{M}.

For an arbitrary point p_{0} in R_{M}^{∗} − R_{ϵ}, there exists a non-trivial simple closed curve C_{p_{0}} passing through p_{0} with ℓ(C_{p_{0}}) < M. Replacing M > 0 so large if necessary, we may assume that C_{p_{0}} is not homotopic to a puncture(Proposition 1). Then there exists a simple closed geodesic L_{p_{0}} which is homotopic to C_{p_{0}}. The length of L_{p_{0}} is greater than ϵ. Hence there exists a constant B depending only on ϵ and M such that d_{ℍ}(p_{0}, L_{p_{0}}) ≤ B. This implies that for every z_{0} ∈ R̃_{M}^{∗} which is not projected to R_{ϵ}, there is an axis ℓ_{0} of a hyperbolic element of Γ such that d_{ℍ}(z_{0}, ℓ_{0}) ≤ B and that the projection to R is a simple closed geodesic with length less than M.

Here, we consider the following two cases for d_{ℍ}(z_{1}, ℓ_{n}).

1: d_{ℍ}(z_{1}, ℓ_{n}) ≤ 4(B + M + 1). In this case, we set g_{1,n} = γ_{1}, g_{2,n} = γ_{2} and g_{3,n} = γ_{n}. Then the third condition of the lemma holds for D = max(4(B + M + 1), d_{ℍ}(z_{1}, ℓ_{2}^{0})). Other three conditions are trivial from the choice of these transformations.

2: d_{ℍ}(z_{1}, ℓ_{n}) > 4(B + M + 1).

In this case, there are points z′_{n} and z′′_{n} on C_{n} that satisfy d_{ℍ}(z_{n}, z′_{n}) = d_{ℍ}(z′_{n}, z′′_{n}) = 2(B + M + 1). Since z′_{n} and z′′_{n} are points on R̃_{M}^{∗} which are not projected to R_{ϵ}, it follows from the above observation,that there exists an axis ℓ′_{n} (resp. ℓ′′_{n}) of γ′_{n} (resp. γ′′_{n}) in Γ such that d_{ℍ}(z′_{n}, ℓ′_{n}) ≤ B (resp. d_{ℍ}(z′′_{n}, ℓ′′_{n}) ≤ B). Since d_{ℍ}(z′_{n}, z′′_{n}) = 2(B + M + 1), we see that ℓ′_{n} and ℓ′′_{n} are distinct. Take a point w_{n} ∈ ℓ′_{n} so that d_{ℍ}(z′_{n}, w_{n}) ≤ B.

If χ_{n}(γ′_{n}) = γ′_{n} and χ_{n}(γ′′_{n}) = γ′′_{n}, set g_{1,n} = γ′_{n}, g_{2,n} = γ′′_{n} and g_{3,n} = γ_{n}. Noting that

d_{ℍ}(w_{n}, ℓ′′_{n}) ≤ 2(B + M + 1) + 2B

and

d_{ℍ}(w_{n}, ℓ_{n}) ≤ 2(B + M + 1) + B,

we see that the third condition of the lemma holds for D = 4B + 2(M + 1). If χ_{n}(γ′_{n}) ≠ γ′_{n} or χ_{n}(γ′′_{n}) ≠ γ′′_{n}, replace γ_{n} by γ′_{n} or γ′′_{n}. Repeating this argument, we get desired elements.

We proceed to prove Proposition 4. Using g_{1,n}, g_{2,n} and g_{3,n} obtained in Lemma 7, we have

K(f_{n}) ≥ A = A(M, D) > 1

from Proposition 2. Since constants M and D are independent of n, this contradicts lim_{n→∞} K(f_{n}) = 1. Hence we have proved this proposition.

Proof of Theorem 1. Let p_{0} = [R, id] be the base point of T^{#}(R). We first suppose that there exists a sequence {g_{n}} of quasiconformal self-maps of R which determine distinct elements of Mod_{c}^{#}(R) such that lim_{n→∞} g_{n}(p_{0}) = p for some p in T^{#}(R). Consider the sequence {f′_{n} = g_{n+1}^{−1} ∘ g_{n}}. Then we see that f′_{n}(p_{0}) converges to p_{0}. Thus there exist quasiconformal mappings f_{n} : R → R (n = 1, 2, …) such that f_{n} is RT-equivalent to f′_{n} and that lim_{n→∞} K(f_{n}) = 1. From Proposition 3, there exists a conformal self-map f of R such that [f_{n} ∘ f] ∈ Mod_{c}^{#}(R) and f_{n} ∘ f converge to the identity on R locally uniformly. Since lim_{n→∞} K(f_{n} ∘ f) = lim_{n→∞} K(f_{n}) = 1, it follows from Proposition 4 that [f_{n} ∘ f] = [id] for sufficiently large n. Hence [f_{n}] = [f^{−1}] for sufficiently large n. This contradicts the assumption that all f_{n} are distinct.

Finally, we see that the same argument as above is valid for an arbitrary point q = [S, f] in T^{#}(R). To see this, it suffices to show that the conditions of Theorem 1 is invariant under the quasiconformal deformation. Namely, the following lemma concludes the theorem.

Lemma 8. Let f : R → S be a K-quasiconformal homeomorphism. Suppose that a Riemann surface R satisfies the both conditions in Theorem 1. Then S also satisfies them.

Proof. Let f̃: ℍ → ℍ be a lift of K-quasiconformal map f. The quasiconformal map f̃ can be extended to ℍ ∪ ℝ̂ with f̃(∞) = ∞ and the restriction f̃|ℝ of f̃ to ℝ is a quasisymmetric function. The Douady-Earle extension Φ(f̃) of f̃|ℝ to ℍ is a quasiconformal and bilipschitz map, and the bilipschitz constant K′ depends only on K (cf. [2]). The projection ϕ_{f} : R → S of Φ(f̃) satisfies

(1/K′)ℓ(c) ≤ ℓ(ϕ_{f}(c)) ≤ K′ℓ(c)

for an arbitrary curve c on R, and [S, f] = [S, ϕ_{f}] in T^{#}(R). Then for an arbitrary point a in ϕ_{f}(R_{M}^{∗}), there exists a non-trivial simple closed curve c_{0} containing a such that ℓ(c_{0}) ≤ K′M. Thus, ϕ_{f}(R_{M}^{∗}) ⊂ S_{K′M}. Therefore, we see that the Riemann surface S satisfies the second condition in Theorem 1 for a connected component of S_{K′M} containing ϕ_{f}(R_{M}^{∗}).

The same argument also shows that the first condition is satisfied by S.

Proof of Theorem 2. Suppose that R is a Riemann surface of positive finite genus g and satisfies the conditions in Theorem 1. Further suppose that Mod^{#}(R) is not discrete. Then there exists a sequence {f_{n}} of quasiconformal self-maps of R which determine distinct elements of Mod^{#}(R) such that lim_{n→∞} K(f_{n}) = 1. Let l be a dividing simple closed curve such that one of components of R \ l is a Riemann surface S of genus g with only one boundary component. Take a non-dividing simple closed geodesic c on S. Then f_{n}(c) ∩ S̄ ≠ ∅ for all n. Indeed, if f_{n}(c) ⊂ S̄^{c}, then f_{n}(c) should be a dividing curve. Since c is a non-dividing curve and f_{n} is a homeomorphism, it can not occur. Then from Lemma 5, there exists a subsequence of {f_{n}} which converges to a conformal self-map f of R locally uniformly on R. Hence we can apply Proposition 4, and we conclude a contradiction.

Next suppose that R has finite positive number of cusps and satisfies the conditions in Theorem 1. If Mod^{#}(R) is not discrete, then there exists a sequence {f_{n}} as above. Let V be a cusp neighborhoods of a puncture of R. Since R has only finitely many cusps, we may assume that f_{n}(V)∩V ≠ ∅ for all n by taking a subsequence of {f_{n}}. Let S be a pair of pants in R such that it contains V and that the boundaries of S consist of the puncture and two dividing simple closed geodesics, say c_{1} and c_{2}. We may assume that two geodesics c_{1} and c_{2} are not homotopic to a boundary component of R. If f_{n}(c_{1}) is homotopic to c_{1} for infinity many n, then they determine elements of Mod_{c_{1}}^{#}(R). Hence, they must be discrete from Theorem 1. Assume that f_{n}(c_{1}) is not homotopic to c_{1} for all n. Since f_{n}(V) ∩ V ≠ ∅ and f_{n}(S) is still a pair of pants for each n, we see that f_{n}(c_{1}) ∩ (S̄ \ V) ≠ ∅ or f_{n}(c_{2}) ∩ (S̄ \ V) ≠ ∅. We may assume that f_{n}(c_{1}) ∩ (S̄ \ V) ≠ ∅. Then from Lemma 5 and Proposition 4, we conclude a contradiction.

Finally, suppose that R has finite positive number of borders and satisfies the conditions in Theorem 1. If Mod^{#}(R) is not discrete, then there exists a sequence {f_{n}} as before. Let B be a one of borders of R. Since R has only finite number of borders, we may assume that f_{n}(B) = B for all n. Let c be a simple closed geodesics which is homotopic to B. Then f_{n}(c) is homotopic to c. Thus f_{n} ∈ Mod_{c}^{#}(R), and {f_{n}} is discrete by Theorem 1. This contradicts lim_{n→∞} K(f_{n}) = 1. Hence Mod^{#}(R) is discrete.

6. Further examples

In Example 4, we showed that there exists a Riemann surface R that satisfies the two conditions in Theorem 1, but that Mod^{#}(R) is not discrete. In this case, there exists a sequence {ω_{n}} of distinct elements of Mod^{#}(R) such that ω_{n}(p_{0}) = p_{0} for any n, where p_{0} = [R, id] ∈ T^{#}(R) . By modifying this example, we see that the following example gives a Riemann surfaces R that there exists a sequence {ω_{n}} of distinct elements of Mod^{#}(R) such that lim_{n→∞} d_{T} (ω_{n}(p), p) = 0 for some p ∈ T^{#}(R) and ω_{n}(p) ≠ p for any n.

Example 5. There is a Riemann surface R without cusps that satisfies two conditions in Theorem 1, but that Mod^{#}(R) is not discrete.

We will give a sketch of a construction of such an R. Consider a torus A_{0} with two geodesic borders of the same length. Let B_{0} be another torus obtained via the (1 + ϵ_{0}) quasiconformal deformation of A_{0} for some ϵ_{0} > 0. Attach two copies of B_{0} to A_{0} along the borders suitably, and we obtain a Riemann surface A_{1}. Hence, it is a Riemann surface of genus 3 with two geodesic borders.

Next we take a Riemann surface B_{1} which is the (1 + ϵ_{1}) quasiconformal deformation of A_{1} for some ϵ_{1} > 0. Attach two copies of B_{1} to A_{1} along the borders suitably, and we obtain a Riemann surface A_{2} which is a Riemann surface of genus 9 with two geodesic borders. Repeating this process, we have a sequence of Riemann surfaces {A_{n}} for some sequence {ϵ_{n}} of positive numbers. We denote the inductive limit of these A_{n} by R. If the sequence {∑_{n=1}^{p} ϵ_{n}}_{p=1}^{∞} is bounded, then we see that R satisfies the above conditions on the injectivity radius.

Let f_{n} be a quasiconformal selfmap of R which sends a part corresponding to A_{n} to a part corresponding to B_{n}. If we take a sequence {ϵ_{n}} converges to zero so rapidly, then we verify that the maximal dilatations of f_{n} converge to 1. Thus, f_{n} induces an element of Mod^{#}(R) whose orbits of p_{0} = [R, id] converge to p_{0} in T^{#}(R).

Next, we modify the above construction to obtain another kind of examples as follows.

Example 6. We give a planar surface R with no short geodesics such that Mod^{#}(R) is not discrete.

For instance, set

z_{n} = \{ \begin{matrix}n + \frac{\sqrt{−1}}{j(n)+1}, & (n ≠ 0) \\ 0, & (n = 0) \end{matrix}

where j(n) is the power of the factor 2 when we decompose |n| to the product of primes. And set R = ℂ − ∑_{n=−∞}^{∞}{z_{n}}.

Now for every positive m, we take a locally affine quasiconformal selfmap f_{m} of R such that Re f_{m}(z) = Re z + 2^{m} (and hence f_{m}(z_{n}) = z_{(n+2^{m})}). Then, since j(n + 2^{m}) = j(n) if j(n) < m, we may take f_{m} so that the maximal dilatations of f_{m} tend to 1. Hence Mod^{#}(R) is not discrete.

We shall construct a Riemann surface R and sequences {M_{n}}, {M′_{n}} having the properties referred in Remark 3 in §4.

Example 7. We consider right-angled hexagons H_{n} (n = 1, 2, …) in the hyperbolic plain ℍ. The sides of the hexagon H_{n} are labelled a_{j,n} (j = 1, 2, …, 6) counterclockwise. We construct the hexagon so that ℓ(a_{2,n}) = ℓ(a_{6,n}), ℓ(a_{3,n}) = ℓ(a_{5,n}) = 1 and ℓ(a_{1,n}) = (2n)^{−1}. Then {H_{n}} converges to a pentagon with a cusp as n → ∞. Thus, we see that

d_{ℍ}(P_{n}, a_{2,n}) = d_{ℍ}(P_{n}, a_{6,n}) ≤ M < ∞ (3)

holds for some M independent of n, where P_{n} is the midpoint of a_{4,n}. Take the perpendicular line L_{j,n} (j = 2, 6) from P_{n} to a_{j,n}. As n → ∞, we see that d_{ℍ}(a_{1,n}, L_{2,n}) = d_{ℍ}(a_{1,n}, L_{6,n}) → ∞.

Now, we take k(n) copies of H_{n}, say H_{n}^{1}, …, H_{n}^{k(n)}, so that

\frac{1}{3}d_{ℍ}(a_{1,n}, L_{2,n}) ≤ 2k(n)ℓ(a_{1,n}) = \frac{1}{n}k(n) ≤ \frac{1}{2}d_{ℍ}(a_{1,n}, L_{2,n}). (4)

Obviously, k(n)/n → ∞ as n → ∞. We denote the sides of H_{n}^{i} corresponding to a_{j,n} by a_{j,n}^{i} (i = 1, 2, … k(n); j = 1, 2, …, 6) and glue H_{n}^{i} and H_{n}^{i+1} along a_{6,n}^{i} and a_{2,n}^{i+1}. Then, we have a right-angled (2k(n) + 4)-gon D_{n} in ℍ. Label the side of D_{n} formed by a_{1,n}^{1} ∪ … ∪ a_{1,n}^{k(n)} b_{1,n} and denote the rest of sides by b_{2,n}, …, b_{2k(n)+4,n} counterclockwise.

We take a copy of D′_{n} of D_{n} with sides b′_{j,n} (j = 1, 2, …, 2k(n) + 4) corresponding to b_{j,n} of D_{n}. We glue D_{n} and D′_{n} along b_{j,n} and b′_{2k(n)+6−j,n} for j = 2, 4, …, 2k(n) + 2 and 2k(n) + 4. Then we have a hyperbolic bordered surface S_{n} of type (0, k(n)+1). The boundary ∂S_{n} consists of one long curve c_{1,n} and k(n) short curves c_{2,n}, …, c_{k(n),n}. It follows from the construction that

ℓ(c_{1,n}) = \frac{k(n)}{n},

ℓ(c_{2,n}) = ℓ(c_{k(n),n}) = 2,

and

ℓ(c_{3,n}) = … = ℓ(c_{k(n)−1,n}) = 4.

From (3), we verify that (S_{n})_{4M} is connected and the naturel map of π_{1}((S_{n})_{4M}) to π_{1}(S_{n}) is surjective. On the other hand, it follows from (4) that (S_{n})_{k(n)/n} is not connected while both (S_{n})_{k(n)/2n} and (S_{n})_{2k(n)/n+4M} are connected.

We take a sequence {j_{n}} so that

4M < \frac{k(j_{n})}{j_{n}} < \frac{k(j_{n+1})}{10j_{n+1}}. (n = 1, 2, …)

We glue S_{j_{n}} and S_{j_{n+1}} along c_{k(j_{n}),j_{n}} of ∂S_{j_{n}} and c_{2,j_{n+1}} of ∂S_{j_{n+1}}. Then we have a bordered Riemann surface S, and a Riemann surface R whose convex core is S. From the construction we verify that R_{M_{n}} is connected for M_{n} = k(j_{n})/2j_{n} but R_{M′_{n}} is not connected for M′_{n} = k(j_{n})/j_{n}. Since M_{n}, M′_{n} > 4M, the natural maps of π_{1}(R_{M_{n}}) and π_{1}(R_{M′_{n}}) to π_{1}(R) is surjective. Thus, R, {M_{n}} and {M′_{n}} are our desired ones.

References

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Ege Fujikawa Department of Mathematics, Tokyo Institute of Technology Oh-okayama Meguro-ku Tokyo 152-8551 Japan

fujikawa@math.titech.ac.jp

Hiroshige Shiga Department of Mathematics, Tokyo Institute of Technology Oh-okayama Meguro-ku Tokyo 152-8551 Japan

shiga@math.titech.ac.jp

Masahiko Taniguchi Department of Mathematics, Graduate School of Science, Kyoto University Kitashirakawa Sakyo-ku Kyoto 606-8502 Japan

tanig@kusm.kyoto-u.ac.jp