Action Of Mod Research Articles

A lower bound of maximal dilatations of quasiconformal automorphisms acting on Fuchsian groups

Let H be the upper half-plane equipped with the hyperbolic metric |dz|/Imz, and Γ a Fuchsian group acting on H. A subset S ⊂ H is said to be precisely invariant under a subgroup ΓS of Γ if γ(S) = S for all γ ∈ ΓS and γ(S)∩S = ∅ for all γ ∈ Γ−ΓS. Furthermore a hyperbolic element γ ∈ Γ is said to be simple if the axis of γ is precisely invariant under the cyclic subgroup 〈γ〉 generated by γ. The hyperbolic distance on H is denoted by d. For a quasiconformal automorphism f of H, the maximal dilatation of f is denoted by K(f). Let T (Γ) be the Teichmuller space of Γ and Mod(Γ) the Teichmuller modular group of Γ. It is known that if Γ is finitely generated of the first kind, then T (Γ) is finite dimensional and the action of Mod(Γ) on T (Γ) is properly discontinuous. This means that for every sequence {fn}n=1 of quasiconformal automorphisms of H satisfying fn ◦ Γ ◦ f−1 n = Γ and limn→∞K(fn) = 1, there exist an integer N and a conformal automorphism f of H such that fn are coincident with f on the real axis R for all n ≥ N . On the other hand, if Γ is infinitely generated, then T (Γ) is infinite dimensional and the action of Mod(Γ) is not properly discontinuous, in general. This means that there exists a sequence {fn}n=1 of quasiconformal automorphisms fn of H such that fn ◦ Γ ◦ f−1 n = Γ and limn→∞K(fn) = 1. On the basis of this fact, in [3], we proved that if Γ satisfies a certain bound condition on translation length, then Mod(Γ) acts properly discontinuously. The following proposition, which gives a lower bound of the maximal dilatation of a quasiconformal automorphism, is crucial for the proof.

Roots of the canonical bundle of the universal Teichm�ller curve and certain subgroups of the mapping class group

1.1. Throughout this paper, let S be a fixed, smooth (C ~) orientable closed surface of genus g > 2. Let S be equipped with a preferred complex structure; we denote the Riemann surface byX0, and the Teichmtiller space ofX o by T 0. There is a Fuchsian group Fg operating in the unit disk so that X o = U/F o. The universal Teichmiiller curve V o is a fibre space over Teichmtiller space, with Xt, the fibre over tE TO conformally equivalent to the Riemann surface represented by te T 0. (Each X t is diffeomorphic to S.) V o is a complex manifold of dimension 3 9 2 , and we denote its canonical line bundle by K(V0). We define L, a holomorphic complex line bundle over Vo, to be an n th root of K(V0) iffL | K(V0). The purpose of this paper is to study the n ttt roots of K(V0); in particular, we investigate an action of the Teichmtiller modular group Mod(Fg) (= mapping class group) on the (finite) set of n th roots. Denote the tangent and cotangent bundles of a manifold M by T(M) and T*(M), respectively; To(M) and T~(M) denote the corresponding bundles with their zero sections removed. David Mumford observed (informal communication) that over a single Riemann surface X, the n th roots correspond to certain homomorphisms 2: HI(To(X), Z.)~Z.. He suggested trying to work out the action of Mod(F 0) in terms of its action on these homomorphisms. 1.2. Let f : S ~ S be a diffeomorphism, and ] its equivalence class in Mod(Fg). Mod(Fo) acts on V 0 as a group of biholomorphic maps, and that action induces (by pullback) an action on the set of holomorphic complex line bundles which a r e n th roots of K(V0). The n tla roots are a finite set of order n 2g. The action of Mod(Fg) on that set gives a homomorphism Mod(Fg)--*Perm(n2~ where Perm(n 2g) is the permutation group on a set of order n 2g. Let Go, . denote the kernel of that homomorphism, that is, the subgroup of Mod(Fo) which acts trivially on all n tla roots. These groups are of particular interest, because they are normal subgroups of finite index in Mod(Fg). The study of the action of Mod(Fg) o n n th roots leads to the following characterization of these subgroups :

The asymptotic geometry of teichmuller space

INTRODUCTION TEICHMULLER SPACE is the space of conformal structures on a topological surface MR of genus g where two are equivalent if there is a conformal map between them which is homotopic to the identity. This space will be denoted by Tg. Teichmuller proved that when g 12 Tg is homeomorphic to an open 6g-6 dimensional ball. Moreover, his proof showed that this homeomorphism could be realized by the radial map along geodesic rays from a fixed base point. In particular, this homeomorphism gives a natural way to compactify TR by putting the endpoints on the rays. We denote the resulting closed 6g-6 dimensional ball by Fg. The immediate question to ask is to what extent i’g depends on the base point from which Tg was compactified. In this paper we show that the geometry along certain rays depends strongly on their base points. Any diffeomorphism of MB induces an isometry of T,. The group of isometries of Tg induced by the group of diffeomorphisms of MB is called the modular group and is denoted by Mod(g). Since F* is defined in terms of geodesics, a natural question to pose is whether or not the action of Mod (g) extends continuously to Tr There are several reasons to be interested in this questions. First, a continuous map on a closed ball is easier to understand than one on an open ball since it always has a fixed point. This fact has been successfully used by Thurston who compactified Tg in such a way that the action of Mod (g) extended continuously. By examining the fixed points of elements of Mod(g), he gave a geometric description of a canonical element of each connected component of Diff M. Thurston’s compactification, TgT, is also homeomorphic to a closed 6g-6 dimensional ball so it is reasonable to ask if his and Teichmuller’s compactifications are the same; i.e., whether the identity map on the interiors extends to a homeomorphism from ?=g to TgT. There was some evidence that this was true. (See [5] and Theorem 3 below.) However we show (Theorem 2) that the compactifications are distinct. Secondly, compactification by geodesic rays have been used extensively by Mostow (and by numerous others) to study complete hyperbolic manifolds. The covering translations of such a manifold, acting on its universal cover, hyperbolic n-space, H”, extend to the closure R”. The boundary sphere of 8” (the “sphere at infinity”) is naturally identified with the space of rays through any interior point of H”. By studying the action of the fundamental group on the sphere at infinity, Mostow proved his well-known rigidity theorem. The allusion to hyperbolic manifolds is not pure whimsy; the Teichmuller space for the torus is isometric to HZ and Mod.(g) (which is isomorphic to SL(2, Z)) extends continuously to its closure. Moreover, T, was thought to have negative curvature for several years. However, Linch[8] found a mistake in the proof and Masur[9] later showed that T, is, in fact, not negatively curved. Thus the question of the extension of Mod (g) to Tg can be thought of both as a question of generalizing a result which is