On Basis-Conjugating Automorphisms of Free Groups

Let X = { x 1 , … x n } be a free generating set of the free group F n and let H be the subgroup of Aut F n consisting of those automorphisms α such that α( x i ) is conjugate to x i for each i = 1, 2 , …, n . We call H the Z -conjugating subgroup of Aut F n . In [ 1 ] Humphries found a generating set for the isomorphic copy H 1 of H consisting of Nielsen transformations where each is conjugate to u i (see remark 1 below). The purpose of this paper is to find a presentation of H (and hence of H 1 ). Let i ≠ j be elements of {1, 2, …, n }. We denote by ( x i ; x j ) the automorphism of F n which sends x i to and fixes x k if k ≠ i . Let S be the set of all such automorphisms.