Centralizers of automorphisms permuting free generators

Olga Macedońska,Witold Tomaszewski

doi:10.1515/math-2019-0007

Abstract

Abstract By σ ∈ Skm we denote a permutation of the cycle-type km and also the induced automorphism permuting subscripts of free generators in the free group Fkm. It is known that the centralizer of the permutation σ in Skm is isomorphic to a wreath product Zk ≀ Sm and is generated by its two subgroups: the first one is isomorphic to $\begin{array}{} \displaystyle Z_k^m \end{array}$, the direct product of m cyclic groups of order k, and the second one is Sm. We show that the centralizer of the automorphism σ ∈ Aut(Fkm) is generated by its subgroups isomorphic to $\begin{array}{} \displaystyle Z_k^m \end{array}$ and Aut(Fm).

Highlights

This paper was inspired by a question of Vitaly Sushchanskyy who asked about the structure of centralizers of automorphisms permuting free generators in a free group.Another motivation of the present paper are numerous papers in which the authors investigate centralizers of nite subgroups of both Aut(Fn) and Out(Fn)
By σ ∈ Skm we denote a permutation of the cycle-type km and the induced automorphism permuting subscripts of free generators in the free group Fkm
It is known that the centralizer of the permutation σ in Skm is isomorphic to a wreath product Zk Sm and is generated by its two subgroups: the rst one is isomorphic to Zkm, the direct product of m cyclic groups of order k, and the second one is Sm

Summary

Introduction

This paper was inspired by a question of Vitaly Sushchanskyy who asked about the structure of centralizers of automorphisms permuting free generators in a free group Another motivation of the present paper are numerous papers in which the authors investigate centralizers of nite subgroups of both Aut(Fn) and Out(Fn). If an automorphism of Fn is induced by a permutation σ of the cycle-type km (m cycles of length k), the centralizer of σ in Aut(Fn) is nite if and only if σ is a long cycle (e.g. it is the cycle without xed points). Pfa study in [4] a centralizer of a subgroup of Out(Fn) cyclically generated by lone axis fully irreducible outer automorphisms The centralizer of such group is in nite cyclic group. It follows from the main theorem of the present paper that the centralizer of

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Then x ασ i

The Main Theorem