All automorphisms of free groups with maximal rank fixed subgroups

Abstract

The Scott Conjecture, proven by Bestvina and Handel [2] says that an automorphism of a free group of ranknhas fixed subgroup of rank at mostn. We characterise in Theorem A below those automorphisms that realise this maximum. It follows from this characterisation, for example, that any such automorphism has linear growth. In our paper [3], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [4].