Introduction

This article will describe some recent progress on the structure of the group Γg of mapping classes for a compact surface Sg of genus g ≥ 2. A mapping-class is an isotopy class of homeomorphisms (usually assumed to be C∞ diffeomorphisms); occasionally it will be convenient to use the alternative definition of it, valid by virtue of Nielsen's theorem, as an element of the outer automorphism group of the fundamental group Π1(Sg).

I shall not attempt here to catalogue the many ways in which these groups impinge on various parts of mathematics, nor will their properties be developed comprehensively. My concern is with two aspects of the theory, which bear a close relationship to each other. One of them is the purely combinatorial study of how Γg operates on the space of simple loops in Sg, and the other is the geometric action as the Teichmüller modular group on the classifying space Tg = T(Sg) of complex structures on the surface Sg. It transpires that in attempting to analyse the boundary structure of T(Sg) and the extended action of Γg on it, one is naturally led to the former question.

My primary aim in the description of Teichmüller space (§ § 3, 4) which forms the basis of this account has been to provide sufficient background to understand the geometric formulation of Thurston's recent theorem on classification of mapping-classes, in terms of both their action on T(Sg) and the dynamical systems determined by them on Sg.