End invariants of the group of outer automorphisms of a free group
type: Item
instance of: scholarly article
title: End invariants of the group of outer automorphisms of a free group
zbMATH Open document ID: 0833.20044
DOI: 10.1016/0040-9383(94)00042-J
author: Karen Vogtmann
published in: Topology
publication date: 1996-03-12T00:00:00Z
review text: It is known that every finitely generated group has 0, 1, 2 or infinitely many ends. If the group is finitely presented, then the number of ends is equal to the number of ends of any simply connected complex on which the group acts freely with finite quotient. If the ends are simply connected the group is said to be simply connected at infinity (a property which is independent of the complex). The author considers the group \(\text{Out}(F_n)\) of outer automorphisms of a free group of rank \(n\). The author proves that this group has infinitely many ends for \(n=2\) and one end for \(n\geq 3\), but the main result of the paper is that \(\text{Out}(F_n)\) is simply connected at infinity for \(n\geq 5\). The proof is carried out by considering the ``outer space'' \(X_n\) of homothety classes of free minimal actions of \(F_n\) on simplicial \(\mathbb{R}\)-trees, defined in a paper by \textit{M. Culler} and \textit{K. Vogtmann} [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)].
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 794438
zbMATH Keywords: finitely generated groups
zbMATH Keywords: finitely presented groups
zbMATH Keywords: number of ends
zbMATH Keywords: simply connected complex
zbMATH Keywords: outer automorphisms
zbMATH Keywords: free groups
zbMATH Keywords: simply connected at infinity
zbMATH Keywords: free minimal actions
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)00042-j
OpenAlex ID: W2013224725
description: scientific article; zbMATH DE number 794438