On Basis-Conjugating Automorphisms of Free Groups

Abstract

Let X = {x1, … xn} be a free generating set of the free group Fn and let H be the subgroup of Aut Fn consisting of those automorphisms α such that α(xi) is conjugate to xi for each i = 1, 2 , …, n. We call H the Z-conjugating subgroup of Aut Fn. In [1] Humphries found a generating set for the isomorphic copy H1 of H consisting of Nielsen transformationswhere each is conjugate to ui (see remark 1 below). The purpose of this paper is to find a presentation of H (and hence of H1).Let i ≠ j be elements of {1, 2, …, n}. We denote by (xi; xj) the automorphism of Fn which sends xi to and fixes xk if k ≠ i. Let S be the set of all such automorphisms.