Automorphisms of Free Groups Have Finitely Generated Fixed Point Sets

An automorphism of a finitely generated free group F extends to a homeomorphism of the end completion E of F. The set of fixed points of this homeomorphism is finitely generated in a certain sense. In particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense. The emphasis on E instead of F in this paper is analogous to Thurston’s study of surface groups where measured laminations are studied instead of simple closed curves. The techniques in this paper arose from a study of the behaviour of an automorphism under iteration, that is, the dynamics of the automorphism; see [ 11. I thank Bill Thurston for explaining some of his ideas to me for analysing automorphisms of free groups. In particular the result on bounded cancellation below is due to Thurston and Matt Grayson. Peter Scott asked whether the fixed subgroup is always finitely generated in [3], and Gersten first answered this question in [2] using quite different methods.