Coarse median spaces and groups.
type: Item
instance of: scholarly article
title: Coarse median spaces and groups.
zbMATH Open document ID: 1283.20048
published in: Pacific Journal of Mathematics
publication date: 2013-05-29T00:00:00Z
full work available at URL: p04.xhtml
review text: The author introduces and systematically studies `coarse median spaces'. These are metric spaces that admit coarse median structure that can be viewed as a coarse nonpositive curvature condition. The existence of such a median on a geodesic space is quasi-isometry invariant, and so it applies to finitely generated groups via their Cayley graphs. Many naturally occurring spaces and groups admit such structures and the author suggests that it is conceivable that every CAT(0) space does. Following the abstract, the contents of the paper can be briefly described as follows: ``We show that asymptotic cones of coarse median spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded Euclidean plane in the space. Using the centroid construction of \textit{J. A. Behrstock} and \textit{Y. N. Minsky} [J. Lond. Math. Soc., II. Ser. 84, No. 3, 765-784 (2011; Zbl 1269.20032)], we show that the mapping class group has this property, and recover the rank theorem of \textit{J. A. Behrstock} and \textit{Y. N. Minsky} [Ann. Math. (2) 167, No. 3, 1055-1077 (2008; Zbl 1280.57015)] and of \textit{U. Hamenstädt} [preprint, \url{arXiv:math/0512429}]. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.''
reviewed by: Mikhail Belolipetsky
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 6168037
zbMATH Keywords: median algebras
zbMATH Keywords: coarse median metric spaces
zbMATH Keywords: Gromov hyperbolic spaces
zbMATH Keywords: cube complexes
zbMATH Keywords: rank
zbMATH Keywords: mapping class groups
zbMATH Keywords: quasi-isometry invariants
zbMATH Keywords: finitely generated groups
zbMATH Keywords: Cayley graphs
author: B. H. Bowditch
MaRDI profile type: Publication
OpenAlex ID: W2164046245
DOI: 10.2140/PJM.2013.261.53
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description: scientific article; zbMATH DE number 6168037