LIMIT SETS AND REGIONS OF DISCONTINUITY OF TEICHMÜLLER MODULAR GROUPS

EGE FUJIKAWA

Abstract. For a Riemann surface of infinite type, the Teichmüller modular group does not act properly discontinuously on the Teichmüller space, in general. As an analogy to the theory of Kleinian groups, we divide the Teichmüller space into the limit set and the region of discontinuity for the Teichmüller modular group, and observe their properties.

1. Introduction

The Teichmüller modular group Mod(R) for a hyperbolic Riemann surface R whose Fuchsian model is of the first kind is the set of homotopy classes of quasiconformal automorphisms of R, and this is a group of biholomorphic automorphisms of the Teichmüller space T (R). If R is of analytically finite type, it is well known that Mod(R) acts properly discontinuously on T (R). However if R is of infinite type, Mod(R) does not act properly discontinuously on T (R), in general. In [4], we gave a sufficient condition for the proper discontinuity.

In this paper, we introduce new notions, the limit set and the region of discontinuity for a Teichmüller modular group. Actually, we define these notions for the Teichmüller modular group of a general Riemann surface whose Fuchsian model is not necessarily of the first kind. For this purpose, we consider the reduced Teichmüller modular group Mod^{#}(R) acting on a reduced Teichmüller space T^{#}(R), which will be defined in the next section. The limit set Λ(G) for a subgroup G of Mod^{#}(R) is the set of points p in T^{#}(R) such that the orbit of p under G is not discrete, and the region of discontinuity Ω(G) is the complement of the limit set. This is an analogy to the theory of Kleinian groups acting on the Riemann sphere, and we expect that they satisfy similar properties to that of limit sets and regions of discontinuity for Kleinian groups. We prove some of them. However it seems that the essential natures of limit sets and regions of discontinuity for Teichmüller modular groups are different from the case of Kleinian groups. For example, the orbit of a point in a limit set is not dense in the limit set, in general. Hence we have to devise the proofs.

We say that a subgroup G of Mod^{#}(R) is of the first kind if Ω(G) = ∅, and otherwise of the second kind. We show sufficient conditions for Mod^{#}(R) to be of the first kind or of the second kind, and give various examples for each cases.

Acknowledgements. The author would like to express her gratitude to Professors Hiroshige Shiga, Katsuhiko Matsuzaki and Masahiko Taniguchi for their valuable suggestions.

1991 Mathematics Subject Classification. Primary 30F60; Secondly 30C62.

Key words and phrases. infinite dimensional Teichmüller space, Teichmüller modular groups, hyperbolic geometry.

2. Preliminaries

We review theories of Teichmüller spaces and Teichmüller modular groups (cf. [5], [7] and [10] ). Throughout this paper, we assume that a Riemann surface R is hyperbolic. In other words, it is represented by ℍ/Γ for some torsion-free Fuchsian group Γ acting on the upper half-plane ℍ. We also assume that R has the non-abelian fundamental group. In other words, the Fuchsian group Γ is nonelementary. We say that R is of analytically finite type (g, n) if it is a Riemann surface of genus g from which n punctures are removed.

Fix a Riemann surface R. For pairs (S_{i}, f_{i}) of Riemann surfaces S_{i} and quasiconformal maps f_{i} of R onto S_{i}, we say that (S_{1}, f_{1}) and (S_{2}, f_{2}) are 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 by a homotopy that keeps every points of the ideal boundary fixed throughout. The Teichmüller space T (R) with the base Riemann surface R is the set of all the equivalence classes [S, f] of such pairs (S, f) as above. Further we say that (S_{1}, f_{1}) and (S_{2}, f_{2}) are weakly 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 the weakly equivalence classes [S, f] of such pairs (S, f) as above.

We say that two quasiconformal automorphisms 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 the ideal boundary fixed throughout. The Teichmüller modular group Mod(R) is the set of all the equivalence classes [h] of quasiconformal automorphisms h of R. Further we say that two quasiconformal automorphisms h_{1} and h_{2} of R are weakly equivalent if h_{2}^{−1} ∘ h_{1} is homotopic to the identity on R. The reduced Teichmüller modular group Mod^{#}(R) is the set of all the weakly equivalence classes [h] of quasiconformal automorphisms h of R. If R is a Riemann surface whose Fuchsian model is of the first kind, then T^{#}(R) = T (R) and Mod^{#}(R) = 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_{1}, f_{1}], [S_{2}, f_{2}]) = \frac{1}{2} inf_{f_{1}, f_{2}} log K(f_{1} ∘ f_{2}^{−1}),

where K(·) is the maximal dilatation of a quasiconformal map and the infimum is taken over all quasiconformal maps f_{1} and f_{2} determining [S_{1}, f_{1}] and [S_{2}, f_{2}] respectively. It is known that, for any quasiconformal map f of R onto S, there exists a quasiconformal map that has the smallest maximal dilatation in the homotopy class of f. This is called an extremal quasiconformal map.

The reduced Teichmüller space T^{#}(R) is a complete metric space with respect to 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 of Mod^{#}(R) to the automorphism group Aut(T^{#}(R)) of T^{#}(R). With a few exceptional surfaces which do not appear in our present case, the above homomorphism Mod^{#}(R) → Aut(T^{#}(R)) is faithful. This was first proved in [2]. Another proof was given in [3]. Therefore we can identify ω_{∗} with ω and omit the asterisk hereafter.

For a non-trivial simple closed curve c on R, we denote the simple closed geodesic that is homotopic to c by c_{∗}. Then for a quasiconformal map f of R, an inequality

K(f)^{−1}ℓ(c_{∗}) ≤ ℓ(f(c)_{∗}) ≤ K(f)ℓ(c_{∗})

holds, where ℓ(·) is the hyperbolic length of a curve ([12, Lemma 3.1]).

3. The limit set and the region of discontinuity

In this section, as an analogy to the theory of Kleinian groups, we introduce notions of the limit set and the region of discontinuity for a reduced Teichmüller modular group, and investigate these properties. We begin by giving their definitions.

Definition 1. We say that a point p in T^{#}(R) is a limit point for a subgroup G of Mod^{#}(R) if there exist a point q ∈ T^{#}(R) and a sequence {χ_{n}} of distinct elements of G such that lim_{n→∞} d_{T} (χ_{n}(q), p) = 0. The set of the limit points is called the limit set of G, and denoted by Λ(G). The complement T^{#}(R) − Λ(G) of the limit set is denoted by Ω(G), and called the region of discontinuity of G. Similarly, for a subgroup G of the ordinary modular group Mod(R), we can define Λ(G) and Ω(G) in T (R).

For a Riemann surface R of analytically finite type, Λ(Mod(R)) = Λ(Mod^{#}(R)) = ∅. On the other hand, for a Riemann surface R whose Fuchsian model is of the second kind, we always have Ω(Mod(R)) = ∅, since a slight change of the value of a quasiconformal map on the ideal boundary produces a different element of Mod(R). This is the reason why we consider the reduced modular group Mod^{#}(R), not the ordinary modular group Mod(R), for Riemann surfaces R of infinite type. In the next section, we exhibit an example of a Riemann surface R which satisfies both Λ(Mod^{#}(R)) ≠ ∅ and Ω(Mod^{#}(R)) ≠ ∅.

We investigate certain properties of the limit set and the region of discontinuity. There are other equivalent definitions of the limit set.

Lemma 1. For a subgroup G of Mod^{#}(R), let Λ′(G) be the set of points p ∈ T^{#}(R) such that lim_{n→∞} d_{T} (ω_{n}(p), q′) = 0 for a point q′ ∈ T^{#}(R) and a sequence {ω_{n}} of distinct elements of G, and Λ′′(G) the set of points p ∈ T^{#}(R) such that lim_{n→∞} d_{T} (ϕ_{n}(p), p) = 0 for a sequence {ϕ_{n}} of distinct elements of G. Then Λ(G) = Λ′(G) = Λ′′(G).

Proof. For any point p ∈ Λ(G), there exist a point q ∈ T^{#}(R) and a sequence {χ_{n}} of distinct elements of G such that lim_{n→∞} d_{T} (χ_{n}(q), p) = 0. Since the action of Mod^{#}(R) on T^{#}(R) is isometric, we have

lim_{n→∞} d_{T} (q, χ_{n}^{−1}(p)) = lim_{n→∞} d_{T} (χ_{n}(q), p) = 0,

which means that p ∈ Λ′(G).

For any point p ∈ Λ′(G), there exist a point q′ ∈ T^{#}(R) and a sequence {ω_{n}} of distinct elements of G which satisfy lim_{n→∞} d_{T} (ω_{n}(p), q′) = 0. Set ϕ_{n} = ω_{n+1}^{−1} ∘ω_{n}. Then lim_{n→∞} d_{T} (ϕ_{n}(p), p) = 0. If {ϕ_{n}} contains infinitely many distinct elements, then this means that p ∈ Λ′′(G). If {ϕ_{n}} consists of finitely many elements, then there exists an element ϕ such that ϕ_{n} = ϕ for infinitely many n and ϕ(p) = p. We see that ϕ is of infinite order. Indeed, suppose that ϕ is of finite order k ≥ 1. Since ω_{n+1} = ω_{n} ∘ ϕ^{−1}, we have ω_{n+k} = ω_{n} ∘ ϕ^{−k} = ω_{n}. This contradicts that all ω_{n} are distinct. Hence ϕ is of infinite order. Thus ϕ^{n}(p) = p implies that p ∈ Λ′′(G).

Obviously Λ′′(G) ⊂ Λ(G).

First, we show basic properties of the limit set.

Lemma 2. For a subgroup G of Mod^{#}(R), the limit set Λ(G) is G-invariant and closed.

Proof. We may assume that Λ(G) ≠ ∅. Let p ∈ Λ(G) and χ ∈ G. Since p is a limit point, there exists a sequence {χ_{n}} of distinct elements of G which satisfies lim_{n→∞} d_{T} (χ_{n}(p), p) = 0 by Lemma 1. Since χ is an isometry, lim_{n→∞} d_{T} (χ ∘ χ_{n}(p), χ(p)) = 0. Setting ω_{n} = χ ∘ χ_{n}, we have lim_{n→∞} d_{T} (ω_{n}(p), χ(p)) = 0. Thus χ(p) ∈ Λ(G) by Lemma 1, and hence χ(Λ(G)) ⊂ Λ(G). Similarly we have χ^{−1}(Λ(G)) ⊂ Λ(G). Thus χ(Λ(G)) = Λ(G), which means that Λ(G) is G-invariant.

Let {p_{n}} be a sequence of points in Λ(G) that converges to a point p ∈ T^{#}(R). For each p_{n} ∈ Λ(G), we can take a sequence {χ_{n,i}}_{i=1}^{∞} of distinct elements of G such that lim_{i→∞} d_{T} (χ_{n,i}(p_{n}), p_{n}) = 0. For each n, choose i(n) so that d_{T} (χ_{n,i(n)}(p_{n}), p_{n}) ≤ 1/n and that all χ_{n,i(n)} are distinct. Since χ is an isometry, we have

d_{T} (χ_{n,i(n)}(p), p) ≤ d_{T} (χ_{n,i(n)}(p), χ_{n,i(n)}(p_{n}))
+ d_{T} (χ_{n,i(n)}(p_{n}), p_{n}) + d_{T} (p_{n}, p)
≤ 2d_{T} (p, p_{n}) + 1/n.

Then lim_{n→∞} d_{T} (χ_{n,i(n)}(p), p) = 0. Hence p ∈ Λ(G), which means that Λ(G) is closed.

We classify the limit points for a subgroup G of Mod^{#}(R) into three types Λ_{0}(G), Λ_{∞}^{1}(G) and Λ_{∞}^{2}(G) according to their stabilizer.

Definition 2. In a subgroup G of Mod^{#}(R), the stabilizer of a point p ∈ T^{#}(R) is defined by Stab_{G}(p) = {χ ∈ G | χ(p) = p}.

We define Λ_{0}(G) as the set of points p ∈ Λ(G) such that there exists a sequence {χ_{n}} of distinct elements of G that satisfies lim_{n→∞} d_{T} (χ_{n}(p), p) = 0 and that χ_{n}(p) ≠ p for all n, and Λ_{∞}(G) as the set of points p ∈ Λ(G) such that Stab_{G}(p) consists of infinitely many elements. Furthermore we divide Λ_{∞}(G) into two disjoint subsets Λ_{∞}^{1}(G) and Λ_{∞}^{2}(G). The Λ_{∞}^{1}(G) is the set of points p ∈ Λ_{∞}(G) such that there exists an element in Stab_{G}(p) that is of infinite order, and the Λ_{∞}^{2}(G) is the set of points p ∈ Λ_{∞}(G) such that all elements in Stab_{G}(p) are of finite order.

It might be the case that Λ_{0}(G) ∩ Λ_{∞}(G) ≠ ∅.

Lemma 3. The sets Λ_{0}(G), Λ_{∞}^{1}(G) and Λ_{∞}^{2}(G) are G-invariant.

Proof. Let p ∈ Λ_{0}(G). We can take a sequence {χ_{n}}_{n=1}^{∞} of distinct elements of G so that lim_{n→∞} d_{T} (χ_{n}(p), p) = 0 and χ_{n}(p) ≠ p for all n. For any χ ∈ G, we have lim_{n→∞} d_{T} (χ ∘ χ_{n}(p), χ(p)) = 0 and χ ∘ χ_{n}(p) ≠ χ(p), which means that χ(p) ∈ Λ_{0}(G). Similarly, we have χ^{−1}(p) ∈ Λ_{0}(G). Thus Λ_{0}(G) is G-invariant.

The fact that Stab_{G}(χ(p)) = χ∘Stab_{G}(p)∘χ^{−1} implies that the sets Λ_{∞}^{1}(G) and Λ_{∞}^{2}(G) are G-invariant.

We exhibit examples of limit points in Λ_{0}(Mod^{#}(R)).

Example 1. Set

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

By Example 2 in [4], we see that the base point [R, id] belongs to Λ_{0}(Mod^{#}(R)).

Example 2. 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. Set R = ℂ − ∑_{n=−∞}^{∞}{z_{n}}. By Example 6 in [4], we see that the base point [R, id] belongs to Λ_{0}(Mod^{#}(R)).

We also have an example of a limit point in Λ_{∞}^{1}(Mod^{#}(R)).

Example 3. Let R̂ be a compact Riemann surface of genus g ≥ 2, and R a normal covering surface of R̂ whose covering transformation group is a cyclic group ⟨ϕ⟩ generated by a conformal automorphism ϕ of R. Then p_{0} = [R, id] ∈ T^{#}(R) and [ϕ] ∈ Mod^{#}(R) satisfy [ϕ](p_{0}) = p_{0}. Hence p_{0} belongs to Λ_{∞}^{1}(Mod^{#}(R)).

We see that if the limit set has an isolated point, the isolated point belongs to Λ_{∞}^{2}(G). However, we do not know whether the limit set has an isolated point.

Theorem 1. For a subgroup G of Mod^{#}(R), the set Λ(G) − Λ_{∞}^{2}(G) does not have an isolated point.

Proof. We will show that, for any point p ∈ Λ(G) − Λ_{∞}^{2}(G), there exists a sequence {p_{n}} of distinct elements in Λ(G) − Λ_{∞}^{2}(G) such that lim_{n→∞} d_{T} (p_{n}, p) = 0.

If p ∈ Λ_{0}(G), then we can take a sequence {χ_{n}}_{n=1}^{∞} of distinct elements of G so that lim_{n→∞} d_{T} (χ_{n}(p), p) = 0 and χ_{n}(p) ≠ p for all n. It follows from Lemma 3 that χ_{n}(p) belongs to Λ_{0}(G) for all n.

If p = [S, f] ∈ Λ_{∞}^{1}(G), then there exists an element χ = [h] ∈ Stab_{G}(p) such that all χ^{k} are distinct for k ∈ ℕ. Since χ(p) = p, the quasiconformal automorphism f ∘ h^{−1} ∘ f^{−1} of S is homotopic to a conformal map ψ. Note that ψ does not have a fixed point in S. Indeed, if ψ has a fixed point x in S, then x is fixed by infinitely many elements ψ^{k} ∈ Aut(S). However it is known that the action of Aut(S) is properly discontinuous if S has the non-abelian fundamental group ([11, Theorem X. 48]). Thus we have a contradiction. We consider the quotient S/⟨ψ⟩ by the cyclic group ⟨ψ⟩, and denote it by Ŝ. The Riemann surface Ŝ is also hyperbolic type.

First, suppose that Ŝ is not of analytically finite type (0,3). We take a sequence of quasiconformal maps {ĝ_{n}} of Ŝ which are not homotopic to a conformal map on Ŝ for all n and satisfy lim_{n→∞} K(ĝ_{n}) = 1. In particular, a lift g̃_{n} of ĝ_{n} to ℍ is not the restriction of a conformal map on the limit set Λ(Γ̂) of the Fuchsian model Γ̂ of Ŝ. Here we note the following lemma.

Lemma 4. ([9, Lemma 2.22]) For a normal subgroup Γ of a non-elementary Fuchsian group Γ̂, we have Λ(Γ) = Λ(Γ̂) if Γ ≠ {id}. Here Λ(·) means the limit set of a Fuchsian group.

Since S is a normal covering surface of Ŝ, Lemma 4 says that the lift g̃_{n} is not the restriction of a conformal map on Λ(Γ), either. Here Γ is the Fuchsian model of S. Then a lift g_{n} of ĝ_{n} to S is not homotopic to a conformal map on S, and p_{n} = [g_{n}(S), g_{n} ∘ f] ∈ T^{#}(R) is different from p for each n. We also have

lim_{n→∞} d_{T} (p, p_{n}) ≤ lim_{n→∞} log K(g_{n}) = 0.

Since d_{T} (χ^{k}(p_{n}), p_{n}) = log K(g_{n} ∘ ψ^{k} ∘ g_{n}^{−1}) and since g_{n} ∘ ψ^{k} ∘ g_{n}^{−1} (k = 1, 2, ⋯) are distinct conformal automorphisms of g_{n}(S), we see that p_{n} belongs to Λ_{∞}^{1}(G). Thus {p_{n}} is a desired sequence.

If Ŝ is of analytically finite type (0,3), then we consider ⟨ψ^{2}⟩ instead of ⟨ψ⟩. Then S/⟨ψ^{2}⟩ is not of analytically finite type (0,3), and the same proof can be applied.

Corollary 1. For a subgroup G of Mod^{#}(R) such that Λ(G)−Λ_{∞}^{2}(G) is not empty, the limit set Λ(G) is an uncountable set.

Proof. By Theorem 1, the closure \overline{Λ(G) − Λ_{∞}^{2}(G)} of Λ(G) − Λ_{∞}^{2}(G) is a perfect closed set. In a complete metric space, every non-empty perfect closed set is an uncountable set (Cantor. cf. [6, p.156]). Then \overline{Λ(G) − Λ_{∞}^{2}(G)} is an uncountable set. Since \overline{Λ(G) − Λ_{∞}^{2}(G)} ⊂ Λ(G), the limit set Λ(G) is also an uncountable set.

Remark 1. The Riemann surfaces R in Examples 1, 2 and 3 satisfy Λ(Mod^{#}(R))− Λ_{∞}^{2}(Mod^{#}(R)) ≠ ∅.

In the theory of Kleinian groups, it is known that the limit set of a nonelementary Kleinian group Γ coincides with the closure of the loxodromic fixed points of Γ, and it also coincides with the closure of the set of limit points that are not fixed by any elements of Γ. On the analogy of this fact, we propose the following problems.

Problem 1. For a subgroup G of Mod^{#}(R), the set Λ_{0}(G) is dense in Λ(G) − Λ_{∞}^{2}(G). The closure of Λ_{∞}(G) coincides with Λ(G).

Next, we consider the proper discontinuity of G on the region of discontinuity.

Definition 3. We say that a subgroup G ⊂ Mod^{#}(R) acts on a subregion Ω ⊂ T^{#}(R) properly discontinuously if for any p ∈ Ω, there exists a constant r > 0 such that the set {χ ∈ G | χ(B(p, r)) ∩ B(p, r) ≠ ∅} consists of only finitely many elements. Here B(p, r) is an open ball centered at p with radius r.

Proposition 1. Let G be a subgroup of Mod^{#}(R). For any point p in T^{#}(R) − Λ_{0}(G), there exists a constant r > 0 such that χ(B(p, r)) ∩ B(p, r) = ∅ for any χ ∈ G − Stab_{G}(p).

Proof. Suppose that, for any n ∈ ℕ, there exists an element χ_{n} ∈ G − Stab_{G}(p) such that χ_{n}(B(p, 1/n)) ∩ B(p, 1/n) ≠ ∅. We take a point q_{n} ∈ B(p, 1/n) so that χ_{n}(q_{n}) ∈ B(p, 1/n). Since χ_{n} is an isometry, we have

d_{T} (p, χ_{n}(p)) ≤ d_{T} (p, χ_{n}(q_{n})) + d_{T} (χ_{n}(q_{n}), χ_{n}(p)) ≤ 2/n.

Hence lim_{n→∞} d_{T} (p, χ_{n}(p)) = 0. If {χ_{n}} contains infinitely many distinct elements, then p ∈ Λ_{0}(G) since χ_{n}(p) ≠ p for all n. This contradicts p ∉ Λ_{0}(G). If {χ_{n}} consists of finitely many elements, then χ_{n}(p) = p for a sufficiently large n. This contradicts χ_{n} ∉ Stab_{G}(p).

If p ∈ Ω(G), then Stab_{G}(p) consists of only finitely many elements. Thus we have the following corollary.

Corollary 2. Let G be a subgroup of Mod^{#}(R). Then G acts on Ω(G) properly discontinuously.

Remark 2. In general, for a group consisting of isometric transformations acting on a complete metric space, the limit set and the region of discontinuity can be defined as in Definition 1, and they satisfy the same properties as in Lemmas 1, 2, 3 and Proposition 1.

In the last of this section, we propose problems on properties of the limit sets and the regions of discontinuity.

Problem 2. For a subgroup G of Mod^{#}(R) such that Ω(G) is not empty, the limit set Λ(G) is nowhere dense in T^{#}(R).

Problem 3. For a subgroup G of Mod^{#}(R) such that Ω(G) is not empty, the region of discontinuity Ω(G) is connected.

4. Teichmüller modular group of the second kind

In this section, we consider sufficient conditions for Mod^{#}(R) to have a nonempty region of discontinuity. The conditions are given in terms of hyperbolic geometry on R.

Definition 4. For a subgroup G of Mod^{#}(R), we say that G is of the first kind if Ω(G) = ∅, and otherwise of the second kind.

Definition 5. For a constant M > 0, we define R_{M} to be the subset of points p ∈ R such that there exists a non-trivial simple closed curve passing through p whose hyperbolic length 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. Further, a connected component of the ϵ-thin part that corresponds to a puncture is called the cusp neighborhood.

The conditions mentioned above are given as follows.

Definition 6. We say that R satisfies the lower bound condition if there exists a constant ϵ > 0 such that the ϵ-thin part of R consists only of cusp neighborhoods and neighborhoods of geodesics which are homotopic to boundary components. Further we say that R satisfies the upper bound condition if there exist a constant M > 0 and a connected component R_{M}^{∗} of R_{M} such that a homomorphism of π_{1}(R_{M}^{∗}) to π_{1}(R) that is induced by the inclusion map of R_{M}^{∗} into R is surjective.

Remark 3. The lower and upper bound conditions are invariant under quasiconformal deformations ([4, Lemma 8]). In other words, they are regarded as conditions for the Teichmüller space.

The following theorem gives a sufficient condition on Riemann surfaces R for Mod^{#}(R) to be of the first kind.

Theorem 2. If R does not satisfy the lower bound condition, then Mod^{#}(R) is of the first kind.

Proof. Since R does not satisfy the lower bound condition, R has a sequence {c_{n∗}} of distinct non-trivial simple closed geodesics that are not homotopic to boundary components with ℓ(c_{n∗}) → 0 (n → ∞).

Let [h_{n}] be an element of Mod^{#}(R) that is the Dehn twist along c_{n} for each n. By the assumption, we can take a representative h_{n} so that lim_{n→∞} K(h_{n}) = 1. Indeed, the collar C(c_{n}) of c_{n} is conformally equivalent to an annulus A_{n} = {z | 1 < |z| < r_{n}}, and we can take r_{n} so that lim_{n→∞} r_{n} = ∞ by the collar lemma (cf. [8]). Further we can take h_{n} so that h_{n} is the identity on the complement of C(c_{n}) and that the restriction of h_{n} to C(c_{n}) is conjugate to a map h̃_{n} : A_{n} → A_{n} defined by

h̃_{n}(z) = z exp (2πi \frac{log|z|}{log r_{n}} ) .

Then lim_{n→∞} K(h̃_{n}) = 1. Hence lim_{n→∞} d_{T} ([h_{n}](p_{0}), p_{0}) = 0, where p_{0} = [R, id] is the base point of T^{#}(R). This means that p_{0} ∈ Λ(Mod^{#}(R)). Further [h_{n}](p_{0}) ≠ p_{0} implies that p_{0} ∈ Λ_{0}(Mod^{#}(R)).

Let f be an arbitrarily quasiconformal map of R onto S. If f is a K-quasiconformal map, the geodesic f(c_{n})_{∗} that is homotopic to f(c_{n}) satisfies

K^{−1}ℓ(c_{n∗}) ≤ ℓ(f(c_{n})_{∗}) ≤ K ℓ(c_{n∗}).

Then S also has the sequence {f(c_{n})_{∗}} of distinct non-trivial simple closed geodesics with ℓ(f(c_{n})_{∗}) → 0 (n → ∞), and the quasiconformal map f ∘ h_{n}^{−1} ∘ f^{−1} is the Dehn twist along each f(c_{n}). Hence, for any point p = [S, f] ∈ T^{#}(R), we have lim_{n→∞} d_{T} ([h_{n}](p), p) = 0, which means that p ∈ Λ(Mod^{#}(R)). In fact, T^{#}(R) = Λ(Mod^{#}(R)) = Λ_{0}(Mod^{#}(R)).

In [4], we gave a sufficient condition on Riemann surfaces R for some subgroup G of Mod^{#}(R) to satisfy Λ(G) = ∅.

Proposition 2. ([4]) Suppose that R satisfies the lower and upper bound conditions. For a non-trivial simple closed curve c on R, we set

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

Then Λ(Mod_{c}^{#}(R)) = ∅.

Using this result, we have a sufficient condition on Riemann surfaces R for Mod^{#}(R) to be of the second kind.

Theorem 3. If R satisfies the lower and upper bound conditions, then Mod^{#}(R) is of the second kind.

To prove this theorem, we use the following lemma.

Lemma 5. Let R be a Riemann surface satisfying the lower bound condition for a constant ϵ > 0, and c_{0} a non-trivial simple closed geodesic on R. Then there exist a positive constant α < ϵ and a quasiconformal map f of R such that ℓ(f(c_{0})_{∗}) < α and that ℓ(f(c)_{∗}) > α for any other non-trivial simple closed geodesics c ≠ c_{0} on R.

Proof. We take two adjacent pairs of pants P^{j} (j = 1, 2) with three geodesic boundaries c_{0}, c_{1}^{j} and c_{2}^{j} so that the five geodesics are mutually disjoint. We fix j, and denote c_{1} and c_{2} instead of c_{1}^{j} and c_{2}^{j} respectively. Set a_{i} = (1/2)ℓ(c_{i}) and b_{i} = d_{R}(c_{i−1}, c_{i+1}) for i = 0, 1, 2 (if i = 0, then we put i − 1 = 2, and if i = 2, then we put i + 1 = 0). Here d_{R}(·, ·) is the hyperbolic distance on R. Further, set

A_{0} = arccosh (\frac{1 + cosh a_{1} cosh a_{2}}{sinh a_{1} sinh a_{2}} )

and α = min{ϵ/2, A_{0}, b_{1}, b_{2}}. We can take a quasiconformal map f on R so that ℓ(f(c_{0})_{∗}) < α and the restriction of f to the complement of P^{1} ∪ P^{2} is the identity. In particular, f(c_{i}) = c_{i} for i = 1, 2, and f(c) = c for any non-trivial simple closed curve c on R that is not through P^{j} (j = 1, 2). Therefore, from the lower bound condition, ℓ(f(c)_{∗}) > ϵ for such c. Further, for any non-trivial simple closed curve c through P^{j}, we have ℓ(f(c)_{∗}) > α.

Indeed, set a′_{i} = (1/2)ℓ(f(c_{i})_{∗}) and b′_{i} = d_{f(R)}(f(c_{i−1})_{∗}, f(c_{i+1})_{∗}) for i = 0, 1, 2 (if i = 0, then we put i − 1 = 2, and if i = 2, then we put i + 1 = 0). Here d_{f(R)}(·, ·) is the hyperbolic distance on f(R). Then a′_{i} = a_{i} for i = 1, 2. For any non-trivial simple closed curve c through P^{j}, we have ℓ(f(c)_{∗}) > b′_{i} for some i = 0, 1, 2. By the formula for a right-angled hexagon ([1, Theorem 7.19.2]), we have

cosh b_{i}^{(′)} = \frac{cosh a_{i}^{(′)} + cosh a_{i−1}^{(′)} cosh a_{i+1}^{(′)}}{sinh a_{i−1}^{(′)} sinh a_{i+1}^{(′)}} .

Hence b′_{0} > A_{0}. Since a′_{0} < a_{0}, we see that b′_{i} > b_{i} for i = 1, 2. Then, for any non-trivial simple closed curve c through P^{j}, we have ℓ(f(c)_{∗}) > min{A_{0}, b_{1}, b_{2}}. Hence ℓ(f(c)_{∗}) > α.

Proof of Theorem 3: Let f be a quasiconformal map on R obtained by Lemma 5. Setting S = f(R), we will show that the point p = [S, f] ∈ T^{#}(R) belongs to Ω(Mod^{#}(R)). Suppose that p ∈ Λ(Mod^{#}(R)). Then there exists distinct elements [h_{n}] in Mod^{#}(R) such that [h_{n}](p) → p (n → ∞). Let g_{n} be an extremal quasiconformal automorphism of S in the homotopy class of f ∘h_{n}^{−1} ∘f^{−1}. Then K(g_{n}) → 1. Letting α be a constant in Lemma 5, we may assume that K(g_{n}) < α/ℓ(f(c_{0})_{∗}) for all n. Then the geodesic g_{n}(f(c_{0}))_{∗} satisfies ℓ(g_{n}(f(c_{0}))_{∗}) ≤ K(g_{n})ℓ(f(c_{0})_{∗}) < α. By Lemma 5, f(c_{0})_{∗} is the only geodesic on S whose length is less than α. Thus g_{n}(f(c_{0})) is homotopic to f(c_{0}), which implies that [h_{n}] ∈ Mod_{c_{0}}^{#}(R). However this contradicts Proposition 2.

The following proposition gives examples of Riemann surfaces that satisfy the lower and upper bound conditions.

Proposition 3. Let R̂ be an analytically finite Riemann surface, and R a normal covering surface of R̂ which is not a universal cover. Then R satisfies the lower and upper bound conditions.

Proof. The lower bound condition is clearly satisfied. We set R̂_{≥ϵ} = R̂ − R̂_{ϵ}, where R̂_{ϵ} is the ϵ-thin part of R̂. The lift R_{0} of R̂_{≥ϵ} to R is connected and a homomorphism of π_{1}(R_{0}) to π_{1}(R) which is induced by the inclusion map of R_{0} into R is surjective. We will show that R_{0} ⊂ R_{M} for some M > 0. Then the upper bound condition is satisfied for the constant M. Since R is a normal covering surface of R̂ which is not a universal cover, we can take a non-trivial simple closed geodesic ĉ_{∗} on R̂ so that the lifts of ĉ_{∗} to R are non-trivial closed geodesics. For an arbitrary point p ∈ R_{0}, let p̂ be the projection of p. We connect p̂ and ĉ_{∗} by the shortest geodesic ℓ̂. Since R̂_{≥ϵ} is compact, there exists a constant M_{1} such that the hyperbolic length of ℓ̂ is less than M_{1} for all p̂ ∈ R̂_{≥ϵ}. Hence there exists a non-trivial simple closed curve ĉ_{p} through p̂ whose hyperbolic length is less than M = 2M_{1} + M_{2}, where M_{2} is the hyperbolic length of ĉ_{∗}. Considering the lift c_{p} of ĉ_{p} which is through p, we conclude that p ∈ R_{M}.

By Theorem 3 and Proposition 3, we have the following.

Corollary 3. Let R̂ be an analytically finite Riemann surface, and R a normal covering surface of R̂ which is not a universal cover. Then Mod^{#}(R) is of the second kind.

Example 4. Let R be a Riemann surface as in Example 3. Then the base point [R, id] belongs to Λ_{∞}^{1}(Mod^{#}(R)). On the other hand, Mod^{#}(R) is of the second kind by Corollary 3. Thus both Λ(Mod^{#}(R)) ≠ ∅ and Ω(Mod^{#}(R)) ≠ ∅ are satisfied.

We conjecture that the sufficient condition for Mod^{#}(R) to be of the second kind can be weakened as follows. A partial solution will be given in the author’s forthcoming paper.

Conjecture. If R satisfies the lower bound condition, then Mod^{#}(R) is of the second kind. That is, considering Theorem 2, we conjecture that Mod^{#}(R) is of the second kind if and only if R satisfies the lower bound condition.

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Department of Mathematics, Tokyo Institute of Technology, Oh-okayama Meguroku Tokyo 152-8551, Japan

  E-mail address: fujikawa@math.titech.ac.jp