Maximum order of periodic outer automorphisms of a free group
type: Item
instance of: scholarly article
title: Maximum order of periodic outer automorphisms of a free group
zbMATH Open document ID: 0952.20026
author: Zhiqiang Bao
published in: Journal of Algebra
publication date: 2001-01-11T00:00:00Z
review text: Landau proved that the maximum order of an element in the symmetric group \(\Sigma_n\) behaves asymptotically like \(\exp(\sqrt{n\log n})\). In the present paper, it is shown that this Landau type estimate holds also for the maximum order of torsion elements in the outer automorphism group \(\text{Out}F_n\) (automorphisms modulo inner automorphisms) of the free group \(F_n\) of rank \(n\). The proof consists of two parts. First, a complete number theoretical description of the possible orders of elements in \(\text{Out}F_n\) is given, by a careful combinatorial analysis of finite cyclic group actions on finite graphs (on the basis of the result that any finite subgroup of \(\text{Out}F_n\) can be induced by an action of the group on a finite graph with fundamental group \(F_n\)); this characterization is also obtained in a paper of \textit{D. G. Khramtsov} [Algebra Logika 26, No. 3, 376-394 (1987; Zbl 0657.20026)], by different methods. The main result of the present paper is then deduced from this characterization using some number theoretical considerations and the above Landau estimate for the maximal orders in the symmetric groups. In a recent paper of \textit{G. Levitt} and \textit{J.-L. Nicolas} [J. Algebra 208, No. 2, 630-642 (1998; Zbl 0930.11070)], the same Landau type estimate is also obtained for the maximum orders of torsion elements in the general linear groups \(\text{GL}(n,\mathbb{Z})\). (Also submitted to MR).
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 1428249
zbMATH Keywords: torsion elements
zbMATH Keywords: outer automorphism groups
zbMATH Keywords: free groups
zbMATH Keywords: orders of elements
zbMATH Keywords: cyclic group actions on finite graphs
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: jabr.1999.8074
OpenAlex ID: W2010147628
cites work: Q3344126
cites work: Q3762388
cites work: Finite subgroups of groups of outer automorphisms of free groups
cites work: The Maximum Order of an Element of a Finite Symmetric Group
cites work: The maximum order of finite groups of outer automorphisms of free groups
cites work: Über Homoeomorphismen \(n\)-dimensionaler Henkelkoerper und endliche Erweiterungen von Schottky-Gruppen
DOI: 10.1006/JABR.1999.8074
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description: scientific article; zbMATH DE number 1428249