On automorphisms of free groups and free products and their fixed points.

Abstract

Free group outer automorphisms were shown by Bestvina and Randell to have fixed subgroups whose rank is bounded in terms of the rank of the underlying group. We consider the case where this upper bound is achieved and obtain combinatorial results about such outer automorphisms thus extending the work of Collins and Turner. We go on to show that such automorphisms can be represented by certain graph of group isomorphisms called Dehn Twists and also solve the conjuagacy problem in a restricted case, thus reproducing the work of Cohen and Lustig, but with different methods. We rely heavily on the relative train tracks of Bestvina and Randell and in fact go on to use an analogue of these for free product automorphisms developed by Collins and Turner. We prove an index theorem for such automorphisms which counts not only the group elements which are fixed but also the points which are fixed at infinity - the infinite reduced words. Two applications of this theorem are considered, first to irreducible free group automorphisms and then to the action of an automorphism on the boundary of a hyperbolic group. We reduce the problem of counting the number of points fixed on the. boundary to the case where the hyperbolic group is indecomposable and provide an easy application to virtually free groups.