The Automorphism Graph of the Free Group of Rank 2
by
Bilal Khan
City University of New York

In the study of the automorphism group of a free group, J. H. C. Whitehead introduced a graph whose vertices are the elements of F, where two vertices are connected if and only if the corresponding elements of are related by one of a specially chosen set of generators of Aut(F), which have come to be known as the so-called "Whitehead automorphisms". In this paper we consider Whitehead's graph, modulo inner automorphisms and the ``obvious'' automorphisms that are induced by permuting the basis of F-we term this combinatorial object the automorphism graph of the free group. We will use the automorphism graph to answer several questions about the action of Aut(F) on F, and its relationship to the natural length function on F. In particular, we shall characterize the subgraphs of the automorphism graph that are induced by subsets of the form

A(u) = { v F s.t. |v|=|u| and for some Aut(F) v = u } where u is in F.

Here we will consider the case when F has rank 2. We will show that there exist uniform constants C, N, such that for all in u of length at least N, if the subgraph induced by A(u) has more than C vertices then it must be a chain containing at most |u|-5 vertices. This implies that |A(u)| is tightly bounded by 8|u|^2 - 40|u|, resolving a question posed by A. Miasnikov and V. Shpilrain (see the preprint Äutomorphic orbits in free groups" in relation to Open Problem F25 at http://www.grouptheory.org). This structural characterization of the automorphism graph yields an algorithm for testing automorphic conjugacy of elements in F_2 that surpasses the classical Whitehead algorithm.

Date received: February 22, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-82.