Asymptotic hereditary asphericity of metric spaces of asymptotic dimension 1.
type: Item
instance of: scholarly article
title: Asymptotic hereditary asphericity of metric spaces of asymptotic dimension 1.
zbMATH Open document ID: 1255.20042
author: Joanna Zubik
published in: Topology and its Applications
publication date: 2011-01-12T00:00:00Z
review text: \textit{T. Januszkiewicz} and \textit{J. Świątkowski} introduced the concept of asymptotic hereditary asphericity (AHA) [in Geom. Topol. 11, 727-758 (2007; Zbl 1188.20043)]. Using this property, they showed that there exist hyperbolic groups of arbitrary cohomological dimension containing no subgroups isomorphic to fundamental groups of closed nonpositively curved Riemannian manifolds of dimension greater than \(2\). AHA is a quasi-isometry invariant, so it is a group property for finitely generated groups with the word metric. It follows from [loc. cit.] that there exist AHA groups of arbitrary large asymptotic dimension. The result of the paper under review is that all metric spaces with asymptotic dimension \(1\) are AHA. In a preprint Osajda and Świątkowski show that many classes of groups are AHA, and conjecture that all asymptotically aspherical groups of asymptotic dimension \(2\) are AHA. This can be viewed as an analog of the Whitehead conjecture and the paper under review fully examines the case of dimension \(1\).
reviewed by: Gregory C. Bell
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 55M10
zbMATH DE Number: 5835560
zbMATH Keywords: asymptotic hereditary asphericity
zbMATH Keywords: asymptotic dimension
zbMATH Keywords: quasi-isometry invariants
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: simplicial complexes
MaRDI profile type: Publication
full work available at URL: j.topol.2010.08.019
OpenAlex ID: W1985844501
cites work: Q3064156
cites work: The Boundary of Negatively Curved Groups
cites work: Hereditarily aspherical compacta and cell-like maps
cites work: Asymptotic dimension of discrete groups
cites work: Asymptotic dimension of finitely presented groups
cites work: Q4819371
cites work: Simplicial nonpositive curvature
cites work: Filling invariants of systolic complexes and groups.
cites work: On asymptotic dimension of countable Abelian groups.
cites work: On the Asymptotic Homological Dimension of Hyperbolic Groups
DOI: 10.1016/J.TOPOL.2010.08.019
description: scientific article; zbMATH DE number 5835560