On fixed subgroups of maximal rank

Abstract

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fixφ for some φ ∈ Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G ≤ Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic. 1. The fringe of a subgroup Throughout this section let I be an arbitrary non-empty set, and let FI = 〈I | 〉 denote the free group on I. 1.1 Definitions. A graph X = (V, E, ι, τ) consists of two disjoint sets V, E (usually denoted V X and EX) and two maps ι, τ: EX → V X. The elements of V X and EX are called the vertices and edges of X, respectively. The maps ι and τ are the incident maps of X. We consider ι and τ extended to the disjoint union EX ∨ (EX)−1 by setting ιe−1 = τe and τe−1 = ιe, e ∈ EX. The basic example is the I-bouquet, RI = ({∗}, I, ι, τ) where ι and τ are each (necessarily) the constant map