Automorphisms of graph groups.
type: Item
instance of: scholarly article
title: Automorphisms of graph groups.
zbMATH Open document ID: 0682.20022
DOI: 10.1016/0021-8693(89)90319-0
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Given a graph \(\Gamma=(V,E)\), the graph group \(F\langle\Gamma\rangle\) is the group with presentation \(\langle V\mid [E]\rangle\), where \([E]\) denotes the set of commutators \(\{[a,b]\mid\{a,b\}\in E\}\). The graph group \(F\langle\Gamma\rangle\) is modeled to be a group analog of the graph algebra K(\(\Gamma)\) generated as a free associative algebra on \(V\) modulo the ideal generated by commutators \([a,b]=ab-ba\) for \(\{a,b\}\in E\). Graph algebras were first studied by Kim, Makar-Limanov, Neggers and Roush. In this paper the author studies analogs in \(F\langle\Gamma\rangle\) of the Nielsen automorphisms of free groups, which reduce to Nielsen automorphisms in the case \(\Gamma=(V,\emptyset)\). The Centralizer Theorem describes the centralizer of elements \(u\) of \(F\langle\Gamma\rangle\). It follows from this description that the centralizer of a graph group is itself a graph group, which is a very nice result even if expected. A strong conjecture of this paper is that the elementary automorphisms (i.e., the analogs of the Nielsen automorphisms) generate the group of automorphisms of \(F\langle\Gamma\rangle\). The special cases proven are \(\Gamma=T\) a tree and \(\Gamma\) is star two-connected containing no triangles or squares. In arriving at these proofs the author develops a generalization of a theorem due to Humphreys on generating sets of free groups. Several additional useful lemmas and propositions provide further computational insights and demonstrate that a theory of graph groups, just as a theory of graph algebras, has a bright present as well as a bright future.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 4116802
zbMATH Keywords: presentations
zbMATH Keywords: commutators
zbMATH Keywords: Nielsen automorphisms
zbMATH Keywords: Centralizer Theorem
zbMATH Keywords: centralizers of graph groups
zbMATH Keywords: elementary automorphisms
zbMATH Keywords: groups of automorphisms
zbMATH Keywords: generating sets
author: Herman J. Servatius
reviewed by: J. Neggers
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90319-0
OpenAlex ID: W2050007657
cites work: An exact sequence for rings of polynomials in partly commuting indeterminates
cites work: ON WEAKLY DISTINGUISHED BASES AND FREE GENERATING SETS OF FREE GROUPS
cites work: Graph algebras
cites work: Homology of certain algebras defined by graphs
cites work: Q4145882
description: scientific article; zbMATH DE number 4116802