On the Nielsen type and the classification for the mapping class group

Abstract

Let S be a compact surface of genus g with n boundary components, with 2g - 2 + n > 0, and let M(S) be its mapping class group (also known as the Teichmiiller modular group). Using measurd foliations, Thurston [ 15 ] has shown that M(S) contains three different types of elements. Bers has shown this decomposition can be obtained using Teichmiiller distance [3]. Subdividing one of the classes further, he obtains four classes. Nielsen has a series of papers [7; 8, Parts I-III] in which he develops an elaborate theory of fixed points of mapping classes. If t is any homeomorphism of S and h a lift of t (or of a power of 1) to the upper half plane, Nielsen’s theory involves the assignment of a pair of integers to h, called the Nielsen type of h. The purpose of this paper is to show how the Thurston classes and the Bers classes can be defined in terms of the Nielsen types of the lifts of