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

Abstract

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 :