On graphs representing automorphisms of free groups

Abstract

For every automorphism f of the free group F = (xl, ... ,, -) Gersten [2, Theorem 5.6] constructed a finite graph representing f and used this graph to give the first proof of Scott's conjecture that the subgroup of fixed points of f is finitely generated. A simpler construction of a graph representing f is contained in a more general treatment of Stallings [6]; also, as Goldstein and Turner [3] show, this graph can be obtained from Whitehead's 3-dimensional model for f . Apart from the applications in the study of fixed points, graphs representing automorphisms are of some independent interest; Hoare [4] uses them to reprove Whitehead's Cut Vertex lemma [7] and Thurston's Bounded Cancellation theorem [1]. This note offers another construction of Gersten's graph with a very short proof. We say that the graph F represents f if F has the following properties. It is finite with every edge colored blue or red and labeled xi or x-1 ( 1 < i < n), assuming that inverse edges have the same color and inverse labels. Every xl is the label of at least one blue and one red edge. Equally colored edges emanating from the same vertex have different labels. The set of all blue edges, and the set of red edges as well, is the edge set of a maximal tree in F. Finally, there is a vertex * such that for every path p which begins and ends at * one has blue (p) f(red(p)), where blue (p) is the element of F (a word in x obtained by reading out the labels of blue edges while traversing p, and red(p) is defined analogously. (This is a modified definition, an inessential variation of the original one.) Let T be the Cayley graph of F [5, p. 123] and TB and TR copies of T colored blue and red respectively. Let FO be the graph obtained by taking TB and TR together and identifying everyvertex v of TB with the vertex fv -l of TR. It is readily seen that Fo satisfies all conditions of the preceding paragraph except that of being finite. If elements of F are regarded as reduced words in x1 , it is easy to see that a subset A of F containing 1 is the vertex set of a subtree of T if and