On Basis-Conjugating Automorphisms of Free Groups

1986/12/01 by James McCool, J. McCool · 17 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Finite Group Theory Research

paper · pdf · doi:10.4153/cjm-1986-073-3

Abstract

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.

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