The topology at infinity of \(\text{Out} (F_n)\)
type: Item
instance of: scholarly article
title: The topology at infinity of \(\text{Out} (F_n)\)
zbMATH Open document ID: 0954.55011
author: Mladen Bestvina
author: Mark Feighn
published in: Inventiones Mathematicae
publication date: 2001-02-11T00:00:00Z
review text: A locally compact space is said to be \(r\)-connected at infinity provided for every compact set \(K\subset X\) there exists a compact set \(L\subset X\) such that \(L\supset K\) and any map \(\alpha :S^d\to X\smallsetminus L\) with \(d\leq r\) extends to a map \(\widetilde{\alpha}:D^{d+1}\to X\smallsetminus K\). Similarly, a discrete group \(\Gamma\) is said to be \(r\)-connected at infinity if it acts cocompactly, freely and simplicially on an \(r\)-connected simplicial complex \(X\) which is \(r\)-connected at infinity. We note that if \(\Gamma\) acts cocompactly, freely and simplicially on another \(r\)-connected simplicial complex \(Y\), then \(Y\) is necessarily also \(r\)-connected at infinity. There are several natural contractible spaces on which \(\text{Out}(F_n)\) acts, namely Culler-Vogtmann's Outer space and its variants. Outer space is to be regarded as the analog of homogeneous space with the action of an arithmetic group or the Teichmüller space with the action of the mapping class group of a compact surface. In this paper the authors prove that the group \(\text{Out}(F_n)\) is \((2n-5)\)-connected at infinity if \(n\geq 2\). To obtain this result, they construct the bordification of Outer space and develop a Morse theory that applies to (bordified) Outer space in the style introduced by the first author and \textit{N. Brady} [Invent. Math. 129, No. 3, 445-470 (1997; Zbl 0888.20021)]. As a corollary of their analysis of the topology of bordified Outer space they also obtain that \(\text{Out}(F_n)\) is a virtual duality group of dimension \(2n-3\) if \(n\geq 2\).
reviewed by: Kohhei Yamaguchi
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1462440
zbMATH Keywords: bordification
zbMATH Keywords: virtual duality group
zbMATH Keywords: outer space
zbMATH Keywords: Morse theory
zbMATH Keywords: action
MaRDI profile type: Publication
full work available at URL: s002220000068
OpenAlex ID: W2086584809
DOI: 10.1007/S002220000068
description: scientific article; zbMATH DE number 1462440