On automorphisms of free products

Summary

Introduction

Whitehead [8] represented elements of a free group by embedding surfaces in an orient able 3-manifold and represented the action of an automorphism by changing the surfaces. Combinatorial versions of his results were given by Rapaport [7] and by Higgins and Lyndon [3]. These were further developed by McCool [6] and a graphical version of his results is given in [4]. More recently Whitehead's work was extended by Gersten [2] to subgroups of a free group and by Collins and Zieschang [1] to elements of a free product.

In this paper we prove the Peak Reduction theorem of [1] for subgroups of a free product. The notation used differs slightly from that in [1] for reasons that will be apparent. We use a concept from [2] and an action by automorphisms on coset graphs similar to that in [5] and dual to that in [4]. This action is also similar to some ideas of Wicks [9]. The authors are grateful to Don Collins for useful comments and criticisms.

Preliminaries

Let G be a group with identity e. Let X be a set of generators for G closed under taking inverses. If H is a subgroup of G then the based coset graph (Γ, H) has for vertices the right cosets of H, with H itself being the base vertex, and has for each x in X a directed edge labelled x joining K to Kx for each coset K.