Pure Braids, a New Subgroup of the Mapping Class Group and Nite-type Invariants (new Version) Contents 1 Introduction 3 2 Statement of Results 4 2.1

In the study of the relation between the mapping class group ? g of a surface g of genus g and the theory of nite-type invariants of homology 3-spheres, three subgroups of ? g play a large role. They are the Torelli group T g , the Johnson subgroup K g and a new subgroup L L g , which contains K g , deened by a choice of a Lagrangian subgroup L H 1 ((g). In this work we determine the quotient L L g =K g ? g =K g , in terms of the precise description of ? g =K g given by Johnson and Morita. We also study the lower central series of L L g and K g , using some natural imbeddings of the pure braid group in L L g and the theory of nite-type invariants.