Comput. Methods Funct. TheoryDOI 10.1007/s40315-014-0066-y

Coherence of Limit Points in the Fibers overthe Asymptotic Teichmüller Space

Ege Fujikawa

Received: 1 October 2013 / Revised: 19 December 2013 / Accepted: 21 December 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract We consider the infinite dimensional Teichmüller space of a Riemann sur-face of general type. On the basis of the fact that the action of the quasiconformalmapping class group on the Teichmüller space is not discontinuous, in general, wedivide the Teichmüller space into two disjoint subsets, the limit set and the region ofdiscontinuity, according to the discreteness of the orbit by a subgroup of the quasicon-formal mapping class group. The asymptotic Teichmüller space is a certain quotientspace of the Teichmüller space and there is a natural projection from the Teichmüllerspace to the asymptotic Teichmüller space. We consider the fibers of the projectionover any point in the asymptotic Teichmüller space, and show a coherence of thediscreteness on each fiber in the Teichmüller space.

Keywords Riemann surface · Teichmüller space · Quasiconformal mappingclass group · Teichmüller modular group

Mathematics Subject Classification (2010) Primary 30F60; Secondary 30C62 ·37F30

1 Introduction

For a Riemann surface of finite type, the Teichmüller space is finite dimensional andthe action of the quasiconformal mapping class group is always discontinuous. The

Communicated by Kari Hag.

E. Fujikawa (B)Department of Mathematics, Chiba University,1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japane-mail: fujikawa@math.s.chiba-u.ac.jp

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moduli space, which is the quotient space of the Teichmüller space by the quasicon-formal mapping class group, is well studied. On the other hand, for a Riemann surfaceof general type, the Teichmüller space is infinite dimensional and the action of thequasiconformal mapping class group is not discontinuous, in general. To study thedynamical behavior of the orbit of the action of the quasiconformal mapping classgroup and to investigate the moduli space of a Riemann surface of general type, wehave introduced the limit set and the region of discontinuity on the Teichmüller spaceas an analogy to the theory of Kleinian groups [10]. For a subgroup G of the quasicon-formal mapping class group MCG(R) of a Riemann surface R, the limit set is the set ofall points in the Teichmüller space whose orbit under the action of G has an accumula-tion point, and the region of discontinuity is the largest open subset in the Teichmüllerspace where G acts discontinuously. We divide the Teichmüller space into these twodisjoint subsets. In a series of papers [10,11,13,17,19,33], we have observed severalproperties of the limit set and the region of discontinuity, and obtained conditions fora point in the Teichmüller space to belong to the limit set or to the region of disconti-nuity. However in general, for a given point in the Teichmüller space, it is not easy todetermine whether the point belongs to the limit set or to the region of discontinuity.

In this paper, we consider stationary subgroups of MCG(R), which preserves acompact subsurface of R. The stationary property is a generalization of propertiesheld by the mapping class group of a compact Riemann surface, and has been alreadywell investigated. For example, we consider the quotient space of the Teichmüllerspace by a certain stationary subgroup; then we can regard the quotient space as amoduli space (see [18]). In particular, for a Riemann surface having a certain geomet-ric condition, a stationary subgroup of MCG(R) acts at every point in the Teichmüllerspace discontinuously, namely, the behaviors of the orbits of all points in the Teich-müller space are the same and we can say that a coherence of the discontinuity holds(see Proposition 4.2 in §4.1). Here we shall study more general Riemann surfacesthat do not necessarily satisfy the geometric condition. In this case, the coherence inthe Teichmüller space does not hold, in general, but we observe that a coherence in acertain subspace on the Teichmüller space holds.

Our result is related to work on another Teichmüller space called the asymptoticTeichmüller space. The asymptotic Teichmüller space is a certain quotient space of theTeichmüller space, which has a certain relationship with the moduli space [18]. Thereis a natural projection from the Teichmüller space to the asymptotic Teichmüller space,and we consider the fibers of the projection over any point in the asymptotic Teich-müller space. Then we prove that for every closed stationary subgroup of MCG(R)

and for every fiber in the Teichmüller space, there is a coherence on the discretenessof the orbit.

2 Preliminaries

Throughout this paper, we assume that a Riemann surface R admits a hyperbolicstructure, i.e., R is represented as the quotient space D/H of the hyperbolic planeD by a torsion-free Fuchsian group H . We also assume that R has a non-abelianfundamental group. In other words, H is non-elementary. Furthermore, we alwaysassume that a Riemann surface has no ideal boundary at infinity.

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In this section, we develop the theory of dynamics of Teichmüller modular groupsacting on infinite dimensional Teichmüller spaces. For an analytically finite Riemannsurface, which is a Riemann surface obtained from a compact Riemann surface byremoving at most a finite number of points, the mapping class group and its action onthe Teichmüller space are well known and broadly studied. For an analytically infiniteRiemann surface whose Teichmüller space is infinite dimensional, we also considermapping classes in the quasiconformal category.

2.1 Quasiconformal Homeomorphism

An orientation preserving homeomorphism of a domain D ⊂ C is said to be quasi-conformal if f is absolutely continuous on lines and there exists a constant k with0 ≤ k < 1 such that | fz | ≤ k| fz | (a.e.) on D. It is known that if f is a qua-siconformal homeomorphism, then fz �= 0 (a.e.) on D, and thus we can considera quantity μ f = fz/ fz (a.e.) on D. This μ f is a bounded measurable functionon D and satisfies ||μ f ||∞ < 1. We call μ f the complex dilatation of f . We setK ( f ) = (1 + ||μ f ||∞)/(1 − ||μ f ||∞) ≥ 1 and call this the maximal dilatation off . We can also define a quasiconformal homeomorphism on a Riemann surface. Fordetails of other equivalent definitions and properties of quasiconformal homeomor-phisms, see the monographs by Astala et al. [1], Bojarski et al. [2], Gehring and Hag[23] and Lehto and Virtanen [26].

2.2 Teichmüller Space and Asymptotic Teichmüller Space

In this section, we define the Teichmüller space and the asymptotic Teichmüller spacefor a Riemann surface.

Definition 2.1 The Teichmüller space T (R) of a Riemann surface R is the set of allequivalence classes [ f ] of quasiconformal homeomorphisms f of R. Here we say thattwo quasiconformal homeomorphisms f1 and f2 of R are equivalent if there exists aconformal homeomorphism h : f1(R) → f2(R) such that f −1

2 ◦ h ◦ f1 is homotopicto the identity on R.

The Teichmüller space T (R) can be embedded in the complex Banach space ofall bounded holomorphic quadratic differentials on R−, where R− is the complexconjugate of R. In this way, T (R) is endowed with a complex structure. For details,see [25,34]. When R is analytically finite, T (R) is finite dimensional; otherwise T (R)

is infinite dimensional and not locally compact.

Definition 2.2 The Teichmüller distance between two points [ f1] and [ f2] in T (R) isdefined by

dT ([ f1], [ f2]) = 1

2log K ( f ),

where f is an extremal quasiconformal homeomorphism in the sense that its maximaldilatation K ( f ) is minimal in the homotopy class of f2 ◦ f −1

1 .

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By virtue of a compactness property of quasiconformal homeomorphisms, theTeichmüller distance dT is complete on T (R), and is coincident with the Kobayashidistance on T (R); see [20].

The asymptotic Teichmüller space has been introduced by Gardiner and Sullivan[22] for the unit disc and by Earle et al. [6,7] for an arbitrary Riemann surface.

Definition 2.3 We say that a quasiconformal homeomorphism f of a Riemann surfaceR is asymptotically conformal if, for every ε > 0, there exists a compact subset V ofR such that the maximal dilatation K ( f |R\V ) of the restriction of f to R \ V is lessthan 1 + ε.

Definition 2.4 The asymptotic Teichmüller space AT (R) of a Riemann surface R isthe set of all asymptotic equivalence classes [[ f ]] of quasiconformal homeomorphismsf of R. Here we say that two quasiconformal homeomorphisms f1 and f2 of R areasymptotically equivalent if there exists an asymptotically conformal homeomorphismh : f1(R) → f2(R) such that f −1

2 ◦ h ◦ f1 is homotopic to the identity on R.

Since a conformal homeomorphism is asymptotically conformal, there is a projec-tion π : T (R) → AT (R) that maps each Teichmüller equivalence class [ f ] ∈ T (R)

to the asymptotic Teichmüller equivalence class [[ f ]] ∈ AT (R). The fiber over anyπ(p) ∈ AT (R) is denoted by Tp, which is a closed submanifold of T (R).

The asymptotic Teichmüller space AT (R) has a complex structure such that π isholomorphic; see [8,21]. The asymptotic Teichmüller space AT (R) is of interest onlywhen R is analytically infinite. Otherwise AT (R) is trivial, i.e., it consists of just onepoint. Conversely, if R is analytically infinite, then AT (R) is not trivial. In fact, it isinfinite dimensional and non-separable.

Definition 2.5 For a quasiconformal homeomorphism f of R, the boundary dilatationof f is defined by H∗( f ) = inf K ( f |R\V ), where the infimum is taken over allcompact subsets V of R. Furthermore, for a Teichmüller equivalence class [ f ] ∈ T (R),the boundary dilatation of [ f ] is defined by H([ f ]) = inf H∗( f ′), where the infimumis taken over all elements f ′ ∈ [ f ]. The asymptotic Teichmüller distance between twopoints [[ f1]] and [[ f2]] in AT (R) is defined by

dAT ([[ f1]], [[ f2]]) = 1

2log H([ f2 ◦ f −1

1 ]),

where [ f2 ◦ f −11 ] is the Teichmüller equivalence class of f2 ◦ f −1

1 in T ( f1(R)).

The asymptotic Teichmüller distance dAT is coincident with the quotient distanceinduced from dT , and this is complete on AT (R). For every point [[ f ]] ∈ AT (R), thereexists an asymptotically extremal element f0 ∈ [[ f ]] satisfying H([ f ]) = H∗( f0).This is different from the case of the Teichmüller space in that we do not know yetwhether the asymptotic Teichmüller distance is coincident with the Kobayashi distanceon AT (R) or not.

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2.3 Quasiconformal Mapping Class Group

For a Riemann surface R, let QC(R) be the group of all quasiconformal automorphismsof R.

Definition 2.6 A quasiconformal mapping class of R is the homotopy equivalenceclass [g] of quasiconformal automorphisms g ∈ QC(R), and the quasiconformalmapping class group MCG(R) of R is the group of all quasiconformal mappingclasses of R.

It is known that the mapping class group of an analytically finite Riemann surfaceis finitely generated and in particular countable. However, for almost all analyticallyinfinite Riemann surfaces, the quasiconformal mapping class groups are uncountable.A Riemann surface whose quasiconformal mapping class group is countable wasconstructed in [29].

In this paper, we consider to the following subgroup of the quasiconformal mappingclass group.

Definition 2.7 A subgroup G ⊂ MCG(R) is said to be stationary if there exists acompact subsurface W of R such that g(W ) ∩ W �= ∅ for every representative g ofevery element of G.

The stationary property is a generalization of the property which MCG(R) of acompact Riemann surface R has. A sequence of normalized quasiconformal homeo-morphisms whose maximal dilatations are uniformly bounded is sequentially compactin compact-open topology. The stationary property of quasiconformal mapping classescorresponds to the normalization in this context, and hence such a sequence of qua-siconformal mapping classes also has the compactness property if they are uniformlybounded.

Compact-open topology on the space of all homeomorphic automorphisms of Rinduces a topology on the quasiconformal mapping class group MCG(R). More pre-cisely, we say that a sequence of mapping classes [gn] ∈ MCG(R) converges to amapping class [g] ∈ MCG(R) in the compact-open topology if we can choose rep-resentatives gn ∈ [gn] and g ∈ [g] such that gn converge to g uniformly on eachcompact subset in R.

Definition 2.8 We say that a subgroup G of MCG(R) is closed if it is closed in thecompact-open topology on MCG(R).

We give examples of stationary subgroups and closed subgroups of the quasicon-formal mapping class group.

Example 2.9 (i) The pure mapping class group P(R) of a Riemann surface R is thegroup of all quasiconformal mapping classes [g] ∈ MCG(R) such that g fixesall non-cuspidal ends of R. Here an end is said to be non-cuspidal if it does notcorrespond to a puncture. If R has at least three topological ends, then P(R) isstationary; see [13]. Furthermore P(R) is closed.

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(ii) The stable quasiconformal mapping class group G∞(R) of a Riemann surface Ris the subgroup of MCG(R) consisting of all essentially trivial mapping classes.Here a quasiconformal mapping class [g] ∈ MCG(R) is said to be essentiallytrivial if there exists a topologically finite subsurface Vg of finite area in R suchthat, for each connected component W of R \ Vg , the restriction g|W : W → R ishomotopic to the inclusion map id|W : W ↪→ R. Then G∞(R) is a subgroup ofP(R), and is stationary for a Riemann surface R having at least three topologicalends. However, G∞(R) is not closed in almost all cases.

(iii) For a simple closed geodesic c on a Riemann surface R, let Gc(R) be the subgroupof MCG(R) consisting of all quasiconformal mapping classes [g] such that g(c)is freely homotopic to c. Then Gc(R) is stationary and closed; see [28].

For basic properties of stationary subgroups with respect to their closedness anddiscreteness, see [32, Sect. 2].

2.4 Teichmüller Modular Group

Every element [g] ∈ MCG(R) acts on T (R) from the left in such a way that[g]∗ : [ f ] �→ [ f ◦ g−1]. It is evident from the definition that MCG(R) acts on T (R)

isometrically with respect to the Teichmüller distance. It also acts biholomorphicallyon T (R).

Definition 2.10 Let

ιT : MCG(R) → Aut(T (R))

be the homomorphism given by [g] �→ [g]∗, where Aut(T (R)) denotes the group ofall biholomorphic automorphisms of T (R). The image ιT (MCG(R)) ⊂ Aut(T (R))

is called the Teichmüller modular group for R and denoted by Mod(R). We alsocall an element of Mod(R) a Teichmüller modular transformation. For a subgroupG ⊂ MCG(R), we denote ιT (G) by G∗.

For all Riemann surfaces R of non-exceptional type, the homomorphism ιT isinjective (faithful). This was first proved by Earle et al. [5], and another proof wasgiven by Epstein [9] and Matsuzaki [28]. Here we say that a Riemann surface R is ofexceptional type if R has finite hyperbolic area and satisfies 2g +n ≤ 4, where g is thegenus of R and n is the number of punctures of R. It was a problem to determine whetherthe homomorphism ιT is also surjective, especially for an analytically infinite Riemannsurface. After a series of pioneering works by Royden [35], Earle and Gardiner [4] andLakic [24], the final step of this problem has been solved affirmatively by Markovic[27], and thus we have the following.

Proposition 2.11 For every Riemann surface R of non-exceptional type, we haveMod(R) = Aut(T (R)).

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2.5 Asymptotic Teichmüller Modular Group

The quasiconformal mapping class group MCG(R) acts on T (R) preserving the fibersof the projection π : T (R) → AT (R). This means that [g]∗(Tp) = T[g]∗(p) for anyfiber Tp ⊂ T (R) over π(p) ∈ AT (R) containing p ∈ T (R) and for any [g] ∈MCG(R). From this, the action of every [g] descends on AT (R). Namely, everyelement [g] ∈ MCG(R) induces a automorphism [g]∗∗ of AT (R) by [[ f ]] �→ [[ f ◦g−1]], which is biholomorphic as well as isometric with respect to dAT ; see [7].

Definition 2.12 Let

ιAT : MCG(R) → Aut(AT (R))

be the homomorphism given by [g] �→ [g]∗∗, where Aut(AT (R)) denotes the group ofall biholomorphic automorphisms of AT (R). The image ιAT (MCG(R)) is called theasymptotic Teichmüller modular group for R (or the geometric automorphism groupof AT (R)) and denoted by ModAT (R). We also call an element of ModAT (R) anasymptotic Teichmüller modular transformation. For a subgroup G ⊂ MCG(R), wedenote ιAT (G) by G∗∗.

It is different from the case of the representation ιT that the homomorphism ιAT isnot injective, namely, Ker ιAT �= {[id]} unless R is either the unit disc or the once-punctured disc; see [5].

Definition 2.13 We call an element of Ker ιAT an asymptotically trivial mapping classand Ker ιAT the asymptotically trivial mapping class group.

We give examples of asymptotically trivial mapping classes.

Example 2.14 (i) A Dehn twist along a simple closed geodesic c on a Riemann surfaceR belongs to Ker ιAT , since it is a deformation only on a compact subsurface, thecollar of c.

(ii) Suppose that there exists a sequence of mutually disjoint simple closed geodesics{cn}n∈N on R, and let γ ∈ MCG(R) be a mapping class that is caused by infinitelymany Dehn twists with respect to each cn . If the hyperbolic length �(cn) of cn

tends to 0 as n → ∞, then γ ∈ Ker ιAT . However, if �(cn) are all the same, thenγ /∈ Ker ιAT .

(iii) For every Riemann surface R, every quasiconformal mapping class induced bya conformal automorphism of R of infinite order does not belong to Ker ιAT .If we assume that R satisfies a certain geometric condition (called the boundedgeometry condition, see Sect. 4), then every mapping class induced by a conformalautomorphism (not necessarily of infinite order) does not belong to Ker ιAT ; see[31] and [18, Sect. 8].

The problem corresponding to Proposition 2.11 in the asymptotic Teichmüller spaceis to determine all elements in Ker ιAT . In [18], we obtained a complete topologicalcharacterization of Ker ιAT . In fact, we proved that Ker ιAT is coincident with the stablequasiconformal mapping class group G∞(R) under the bounded geometry conditionof a Riemann surface R.

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2.6 Limit Sets and Regions of Discontinuity

We review the definitions of the limit set and the region of discontinuity for the Teich-müller modular group, which were introduced in [10]. For a subgroup � ⊂ Mod(R),the stabilizer of a point p ∈ T (R) is defined by Stab�(p) = {γ ∈ � | γ (p) = p}.

Definition 2.15 We say that a subgroup � ⊂ Mod(R) acts at a point p ∈ T (R)

discontinuously if the following equivalent conditions are satisfied:

(a) there exists a neighborhood U of p such that the number of elements γ ∈ �

satisfying γ (U ) ∩ U �= ∅ is finite.(b) there exists no sequence of distinct elements {γn}∞n=1 ⊂ � such that

limn→∞ dT (γn(p), p) = 0.

(c) the orbit �(p) is discrete and the stabilizer subgroup Stab�(p) is finite.

For a subgroup � ⊂ Mod(R), we define the region of discontinuity T (�) for � asthe set of all points p ∈ T (R) where � acts discontinuously.

On the other hand, we also define the set of all accumulation points in T (R) of theorbit by the action of a subgroup of Mod(R).

Definition 2.16 For a subgroup � ⊂ Mod(R), it is said that q ∈ T (R) is a limit pointof p ∈ T (R) for � if there exists a sequence {γn}∞n=1 of distinct elements of � suchthat dT (γn(p), q) → 0 as n → ∞. The set of all limit points of p for � is denotedby T (�, p), and the limit set for � is defined by T (�) = ⋃

p∈T (R) T (�, p). Itis said that p ∈ T (R) is a recurrent point for � if p ∈ T (�, p), and the set of allrecurrent points for � is called the recurrent set for � and is denoted by RecT (�).

Then RecT (�) = T (�) and they are closed for every subgroup � ⊂ Mod(R).Furthermore, T (�) = T (R) \ T (�) for every subgroup � ⊂ Mod(R), namely, wedivide the Teichmüller space into the two disjoint subsets.

The limit set is originally defined for a Kleinian group and it is also defined for theiteration of a holomorphic function as the Julia set. Some properties of our limit setare common to the original settings but some are not. For example, the limit set is �-invariant and closed (see [10]). Furthermore, under the bounded geometry conditionof a Riemann surface R, the region of discontinuity T (�) is dense in T (R) andconnected for every subgroup � ⊂ Mod(R) (see [33]). However we do not knowwhether the limit set has an isolated point or not. In general, we can define the limitset and region of discontinuity for the group of isometries acting on a complete metricspace (see [16]).

3 Results

In a couple of papers [10,11], we have obtained conditions for a point to belong tothe limit set or to the region of discontinuity. However, in general, for a given point

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in the Teichmüller space, it is not easy to determine whether the point belongs to thelimit set or to the region of discontinuity. We consider a stationary subgroup of thequasiconformal mapping class group and prove the coherence of the discreteness ofthe orbit of all points on each fiber of the projection π : T (R) → AT (R).

Theorem 3.1 Let R be a Riemann surface, and G a subgroup of MCG(R) that isstationary and closed. Then for every point p ∈ T (R), either Tp ⊂ T (G∗) orTp ⊂ T (G∗) holds.

Proof By changing the base point, it is enough to prove the statement only for the fiberTo containing the origin o = [id] ∈ T (R). More precisely, we assume that o ∈ T (G∗)and prove that p ∈ T (G∗) for an arbitrary point p = [ f ] in the fiber To. Then wemay assume that the representative f is an asymptotically conformal mapping of R.Furthermore by [8] we may assume that f is a quasiconformal diffeomorphism inducedby the barycentric extension given by Douady and Earle [3]. We set S = f (R).

By the assumption o ∈ T (G∗), there exists a sequence {[gn]}∞n=1 of distinctelements of G such that dT ([gn]∗(o), o) → 0 as n → ∞. For every n, we takea representative gn so that it is a quasiconformal diffeomorphism induced by thebarycentric extension. Then K (gn) → 1 as n → ∞ by [3, Cor. 2]. Fix ε > 0arbitrarily. Then there is an integer n0 such that K (gn) < (1 + ε)1/3 for all n > n0.

By the assumption that the subgroup G is stationary, we can take a subsequence of{gn}∞n=1 that converges to a quasiconformal automorphism h of R locally uniformlyon R. Since K (gn) → 1, we may assume that h is conformal. Since we have assumedthat G is closed, the mapping class [h] belongs to G. Set hn = gn ◦ h−1. Then[hn] ∈ G and hn converges to the identity locally uniformly on R. Furthermore,K (hn) = K (gn ◦ h−1) ≤ K (gn) < (1 + ε)1/3 for all n > n0.

We will prove that K ( f ◦ h−1n ◦ f −1) → 1 as n → ∞. Then this yields that

dT ([hn]∗(p), p) → 0 as n → ∞, which implies that p ∈ T (G∗). We divide ourargument into two parts, one is on a certain compact subsurface and the other is outsidethe compact subsurface.

First we consider the maximal dilatation of f ◦h−1n ◦ f −1 outside a certain compact

subsurface of S. Since f is asymptotically conformal, for the fixed ε > 0 above, thereexists a compact subsurface V0 ⊂ R with geodesic boundary such that the maximaldilatation of f restricted to outside V0 satisfies K ( f |R\V0) < (1 + ε)1/3. We takea compact subsurface V with geodesic boundary containing V0 properly. Since hn

converges to the identity locally uniformly on R, we have hn(V ) ⊃ V0 for sufficientlylarge n. Then we see that h−1

n also converges to the identity locally uniformly on R,and we can take an integer n1 ≥ n0 such that h−1

n (R \ V ) ⊂ R \ V0 for all n ≥ n1.Then, for every n ≥ n1, we have

K ( f ◦ h−1n ◦ f −1|S\ f (V )) ≤ K ( f |R\V0)K (h−1

n |R\V )K ( f −1|S\ f (V ))

< 1 + ε.

Next we will show that for every ε > 0, there exists an integer n2 ≥ n1 such thatK ( f ◦ h−1

n ◦ f −1| f (V )) < 1 + ε for all n > n2. For z ∈ S, the complex dilatation

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μ f ◦h−1n ◦ f −1 of the quasiconformal homeomorphism f ◦ h−1

n ◦ f −1 is

μ f ◦h−1n ◦ f −1(z) = fz(w)

fz(w)· μ f ◦h−1

n(w) − μ f (w)

1 − μ f (w)μ f ◦h−1n

(w),

where w = f −1(z) and

μ f ◦h−1n

(w) = (hn)w(h−1n (w))

(hn)w(h−1n (w))

· μ f (h−1n (w)) − μhn (h

−1n (w))

1 − μhn (h−1n (w))μ f (h

−1n (w))

.

Since hn converges to the identity locally uniformly on R, we see that (hn)w con-verges to 1 and (hn)w converges to 0 locally uniformly on R. Indeed, since we haveassumed that R has no ideal boundary at infinity, the limit set of the Fuchsian groupon D representing R is coincident with the unit circle ∂D. A suitable lift hn of hn to D

extends to the boundary ∂D, and the restriction hn|∂D to the boundary ∂D convergesto the identity uniformly; see [30, Prop. 2.1]. Since we have taken gn (and hence hn)as a quasiconformal diffeomorphism induced by the barycentric extension, we canconclude that (hn)w → 1 and (hn)w → 0 locally uniformly by [3, Prop. 2].

We see that μ f ◦h−1n

(w) → μ f (w) locally uniformly on S. Indeed, (hn)w(h−1n (w))

→ 1 locally uniformly and μhn (h−1n (w)) = (hn)w(h−1

n (w))/(hn)w(h−1n (w)) → 0

locally uniformly. Furthermore, since the complex dilatation μ f of the quasicon-formal diffeomorphism f is continuous, we have μ f (h−1

n (w)) → μ f (w). Thenμ f ◦h−1

n(w) → μ f (w), and hence |μ f ◦h−1

n ◦ f −1(z)| → 0 locally uniformly. Sincef (V ) is compact, we obtain ||μ f ◦h−1

n ◦ f −1 | f (V )||∞ → 0 as n → ∞. ��For a non-stationary subgroup of the quasiconformal mapping class group, Theorem

3.1 is not true as the following example says. For a simple closed curve c on a Riemannsurface, let �(c) be the hyperbolic length of the geodesic that is freely homotopic to c.

Example 3.2 Let R be a Riemann surface that admits a conformal automorphism gof infinite order, and let G = 〈[g]〉 ⊂ MCG(R). Then G is not stationary. SinceK (gn) = 1 for every integer n, we have [gn]∗(o) = o for every n and for the basepoint o = [id] ∈ T (R). Thus we conclude that o ∈ T (G∗). On the other hand, let c0be a simple closed geodesic on R, and let f be a quasiconformal homeomorphism of Rsuch that �( f (c0)) = (1/2)�(c0) and �( f (c)) = �(c) for all simple closed geodesics ccontained in R\V . Here V is a sufficiently large compact subsurface containing c0. Thisquasiconformal homeomorphism f is obtained by considering a pants decompositionof R, and f is conformal outside V . In particular, f is asymptotically conformal andthus p := [ f ] ∈ To. However p ∈ T (G∗). Indeed, for the extremal quasiconformalhomeomorphism f ′ ∈ [ f ], we have

dT ([gn](p), p) = K ( f ′ ◦ (gn)−1 ◦ f ′−1) ≥ �( f ′ ◦ (gn)−1 ◦ f ′−1( f ′(c0)))

�( f ′(c0))= 2

for all sufficiently large n.

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4 Remarks

In this section, we give some remarks on Theorem 3.1 by reviewing our previousresults.

4.1 Bounded Geometry Condition

In Theorem 3.1, we do not assume any condition on the Riemann surface. In thissection, we mention a special case of Theorem 3.1 under the following assumption onthe hyperbolic geometry of Riemann surfaces. Hereafter, R denotes the non-cuspidalpart of R obtained by removing all horocyclic cusp neighborhoods of area one. Fora constant M > 0, RM denotes the set of all points x in R satisfying a propertythat there exists a homotopically non-trivial and non-cuspidal closed curve based at xwhose hyperbolic length is less than M .

Definition 4.1 We say that a hyperbolic Riemann surface R satisfies the boundedgeometry condition if the following three conditions are fulfilled:

(i) lower bound condition: there a constant m > 0 such that, for every point x ∈ R,every homotopically non-trivial curve based at x has hyperbolic length greaterthan or equal to m;

(ii) upper bound condition: there exist a constant M > 0 and a connected com-ponent R0

M of RM such that the inclusion map R0M ↪→ R induces a surjective

homomorphism π1(R0M ) → π1(R);

(iii) R has no ideal boundary at infinity.

These conditions are quasiconformally invariant and hence we may regard themas conditions for the Teichmüller space. For example an arbitrary non-universal nor-mal cover of an analytically finite Riemann surface satisfies the bounded geometrycondition (see [10]). If a Riemann surface has a pants decomposition such that all thelengths of boundary geodesics of the pairs of pants are uniformly bounded from aboveand from below, then it satisfies the bounded geometry condition. In particular, thecomplement of the Cantor set in the complex plane satisfies the bounded geometrycondition. However, the converse is not true. Counter examples can be easily obtainedby considering a planar non-universal normal cover of an analytically finite Riemannsurface with a puncture; see [14, Prop. 2.6].

Under the bounded geometry condition, we have already proved the followingstatement in [11], and we can regard Theorem 3.1 as a generalization of Proposition 4.2.

Proposition 4.2 Let R be a Riemann surface satisfying the bounded geometry con-dition. Then T (G∗) = T (R) (namely T (G∗) = ∅) for every stationary subgroupG ⊂ MCG(R). In particular, Tp ⊂ T (G∗) for every point p ∈ T (R).

4.2 Projection to the Asymptotic Teichmüller Space

Similar to the action of the Teichmüller modular group on the Teichmüller space,we also define the region of discontinuity AT (G∗∗) and the limit set AT (G∗∗)

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on the asymptotic Teichmüller space AT (R) for a subgroup G∗∗ of the asymptoticTeichmüller modular group ModAT (R).

On the projection of the limit point in the Teichmüller space to the asymptoticTeichmüller space by π : T (R) → AT (R), we have the following relation.

Proposition 4.3 [15] Let R be a Riemann surface satisfying the bounded geometrycondition. Then π(T (G∗)) ⊂ AT (G∗∗) holds for any subgroup G ⊂ MCG(R).

The proof is based on the fact that Ker ιAT acts on T (R) discontinuously underthe bounded geometry condition. On the other hand, there can exist a point in theregion of discontinuity on the Teichmüller space whose projection to the asymptoticTeichmüller space is in the limit set as the following example shows.

Example 4.4 Let R0 be a normal cover of a compact Riemann surface whose coveringtransformation group is a cyclic group generated by a conformal automorphism of R0of infinite order, and R = R0 \ {p} for a point p ∈ R0. Then MCG(R) is stationary.Indeed, for a compact subsurface W ⊂ R whose boundary consists of two non-trivialdividing simple closed curves and a curve that is homotopic to the puncture p, we haveg(W )∩ W �= ∅ for every g ∈ [g] and every [g] ∈ MCG(R). Then by Proposition 4.2,we haveT (Mod(R)) = T (R). On the other hand, we see thatAT (ModAT (R)) �= ∅;see [12].

Example 4.4 in particular says that even if all points in the fiber Tp ⊂ T (R) belongto the region of discontinuity, the projection π(p) ∈ AT (R) can belong to the limitset. By Theorem 3.1, the coherence of the points in the fiber Tp is satisfied, but thecondition π(p) ∈ AT (G∗∗) does not imply Tp ⊂ T (G∗).

4.3 Non-Stationary Subgroup

By Proposition 4.2, for a Riemann surface R satisfying the bounded geometry con-dition, the stationary property is a sufficient condition for the discontinuous action.However, the stationary property is not a necessary condition. There exist a Riemannsurface R satisfying the bounded geometry condition and a subgroup G of MCG(R)

such that G is non-stationary but G acts on T (R) discontinuously. Furthermore, thereexists a Riemann surface R satisfying the bounded geometry condition such thatMCG(R) is non-stationary but it acts on T (R) discontinuously; see [17].

In general, we have the following result for subgroups of MCG(R) which are notnecessarily stationary.

Proposition 4.5 [10] For a Riemann surface R satisfying the bounded geometrycondition and for a subgroup G ⊂ MCG(R), we have T (G∗) �= ∅.

Corresponding to Example 4.2, the following statement says that even if all pointsin the fiber belong to the region of discontinuity, the projection can belong to the limitset in the asymptotic Teichmüller space for a certain non-stationary subgroup. For anelement [g]∗ ∈ Mod(R), set Fix([g]∗) = {p ∈ T (R) | [g]∗(p) = p}.

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Proposition 4.6 [15] Let R be a Riemann surface. Suppose that R admits a con-formal automorphism g of infinite order such that Fix([g]∗) is not a singleton.Let G = 〈[g]〉 ⊂ MCG(R). Then there is a limit point π(p) ∈ AT (G∗∗) on AT (R)

such that Tp ⊂ T (G∗) on T (R).

By Proposition 4.6, Proposition 4.5 does not imply that AT (G∗∗) �= ∅, but we haveproved that AT (G∗∗) �= ∅ for a certain subgroup G∗∗ ⊂ ModAT (R) by constructinga Riemann surface corresponding to a point in AT (G∗∗); see [15].

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