On the dynamics and the fixed subgroup of a free group automorphism
type: Item
instance of: scholarly article
title: On the dynamics and the fixed subgroup of a free group automorphism
zbMATH Open document ID: 0678.57001
DOI: 10.1007/BF01393699
author: Marshall M. Cohen
author: Martin Lustig
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143690
review text: The study of the automorphism group Aut \(F_ n\) of a free group \(F_ n\) by J. H. C. Whitehead used topological methods up to dimension 4. Later his results were extended and reproven in more algebraic and combinatorial terms. But recently several authors have taken up Whitehead's original ideas again and combined them with Gersten's result that rank Fix(\(\phi)\) \((\phi \in Aut(F_ n))\) is finite, in particular \textit{R. Z. Goldstein} and \textit{E. C. Turner} [Invent. Math. 78, 1-12, (1984; Zbl 0548.20016); Bull. Lond. Math. Soc. 18, 468-470 (1986; Zbl 0576.20016)]. Goldstein and Turner introduced a graph D(\(\phi)\) for \(\phi\in Aut F_ n\), whose vertices correspond to elements of \(F_ n\). Two vertices v, w are connected by an edge, if \(w=x^{-1}v\phi (x)\), where x is an element (or its inverse) of a given standard basis of \(F_ n\). Cohen and Lustig refine this method by orienting all edges of the first barycentric subdivision of D(\(\phi)\); thus the original edges become ``ordinary'', ``attractive'', or ``repulsive''. Studying the dynamics which results from these labels on D(\(\phi)\) (in particular, closed oriented paths and attractive rays towards infinity), they obtain remarkable geometric resp. algebraic formulae for the determination of rank Fix(\(\phi)\). Moreover they deduce new estimates which can be read off in practically given situations, for instance, if \(\phi\) sends each element of the standard basis to a word containing these generators only with positive exponents. As an example, Fix(\(\phi)\) turns out to be trivial for \(\phi\) : F(a,b,c)\(\to F(a,b,c)\) given by \(\phi (a)=b\), \(\phi (b)=c\), \(\phi (c)=ab\), a question raised by Stallings. The material of this paper has already been incorporated in \textit{D. E. Cohen's} new textbook: Combinatorial group theory, a topological approach (LMS, Student Texts, 14) [for a review of an earlier version see Zbl 0389.20024]. It is part of a present broader stream (where also work of Bestvina…
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 4111596
zbMATH Keywords: fixed points of automorphisms of free groups
zbMATH Keywords: automorphism group of a free group
MaRDI profile type: Publication
cites work: Automorphisms of free groups have finitely generated fixed point sets
cites work: Moduli of graphs and automorphisms of free groups
cites work: Fixed points of automorphisms of free groups
cites work: Q3525452
cites work: Automorphisms of the free group of rank two without finite orbits
cites work: Topologically Unrealizable Automorphisms of Free Groups
OpenAlex ID: W2063916249
description: scientific article; zbMATH DE number 4111596