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