On the group of conjugating automorphisms of a free group
type: Item
instance of: scholarly article
title: On the group of conjugating automorphisms of a free group
zbMATH Open document ID: 0902.20016
DOI: 10.1007/BF02308881
author: A. G. Savushkina
published in: Mathematical Notes
publication date: 1998-12-06T00:00:00Z
review text: Denote by \({\mathfrak F}_n\) (\(n\geq 2\)) the free group with free generators \(x_1,\dots,x_n\). An automorphism in the group \(\Aut({\mathfrak F}_n)\) that maps any generator \(x_i\) into a word of the type \(W^{-1}_ix_{p_i}W_i\) (for some word \(W_i=W_i(x_1,\dots,x_n)\) in \({\mathfrak F}_n\)) is called conjugating and the subgroup of all such automorphisms is denoted by \({\mathfrak C}_n\). Those among the conjugating automorphisms given by \(x_i\mapsto W^{-1}_ix_iW_i\), are called basis-conjugating automorphisms; denote their subgroup by \({\mathfrak M}_n\). Further, denote by \({\mathfrak S}_n\) the symmetric group on the set \(X=\{x_1,\dots,x_n\}\) and let \({\mathfrak B}_n\) be the braid group, i.e. the subgroup in \({\mathfrak C}_n\) of all those automorphisms that map the product \(x_1x_2\cdot\ldots\cdot x_n\) into itself. The group \({\mathfrak B}_n\) contains a subgroup called the pure braid group \({\mathfrak N}_n\) and generated by the automorphisms \(\varphi_{i,j}=\psi_{j-1}\psi_{j-2}\cdots\psi_{i+1}\psi^2_i\psi_{i+1}\cdots\psi_{j-2}\psi_{j-1}\) for all \(1\leq i<j\leq n\), with the automorphisms \(\psi_k\) (\(k=1,\dots,n-1\)) given by the rules \(x_k\mapsto x_{k+1}\), \(x_{k+1}\mapsto x^{-1}_{k+1}x_kx_{k+1}\) and satisfying the relations \(\psi_k\psi_{k+1}\psi_k=\psi_{k+1}\psi_k\psi_{k+1}\) (\(k=1,\dots,n-2\)) and \(\psi_k\psi_l=\psi_l\psi_k\) (\(k,l=1,\dots,n-1\); \(| k-l|\geq 2\)). The author proves that the group \({\mathfrak C}_n\) is the semidirect product of the invariant subgroups \({\mathfrak M}_n\) and \({\mathfrak S}_n\). There are given (Theorem 1) a description of the generators of \({\mathfrak C}_n\) via those for \({\mathfrak B}_n\) and \({\mathfrak S}_n\), also expressions for the generators of \({\mathfrak N}_n\) via those for \({\mathfrak M}_n\) (Lemma 3). It is proved that the centers of \({\mathfrak C}_n\) and \({\mathfrak M}_n\) are trivial. Criteria are given (Lemmas 4 and 5) for an automorphism to belong to the commutator subgr…
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1158873
zbMATH Keywords: free groups
zbMATH Keywords: conjugating automorphisms
zbMATH Keywords: bases conjugating automorphisms
zbMATH Keywords: pure braid groups
zbMATH Keywords: generators
zbMATH Keywords: Burau representation
zbMATH Keywords: Gassner representation
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
cites work: On Basis-Conjugating Automorphisms of Free Groups
cites work: ON WEAKLY DISTINGUISHED BASES AND FREE GENERATING SETS OF FREE GROUPS
cites work: Q5650865
cites work: Q5513066
cites work: Braids, Links, and Mapping Class Groups. (AM-82)
cites work: Q5583892
cites work: On braid groups
cites work: On knot groups
full work available at URL: https://doi.org/10.1007/bf02308881
OpenAlex ID: W2067621805
Recommended article: On automorphisms of free groups
Recommended article: On Basis-Conjugating Automorphisms of Free Groups
Recommended article: The problem of conjugacy of automorphisms of free groups
Recommended article: On the conjugacy problem for automorphisms of free groups. With an addendum
Recommended article: On automorphisms of relatively free groups
Recommended article: Automorphisms of automorphism groups of free groups
Recommended article: Automorphisms of Free Groups
Recommended article: On the basis-conjugating automorphism groups of free groups and free metabelian groups
Recommended article: On automorphism-fixed subgroups of a free group
Recommended article: On the Automorphisms of Free Groups and Free Nilpotent Groups
description: scientific article; zbMATH DE number 1158873