Bowditch, B. H. (Brian Hayward), 1961- . (2013) Coarse median spaces and groups. Pacific Journal of Mathematics, Volume 261 (Number 1). pp. 53-93. ISSN 0030-8730

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Abstract

We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. 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. We show that asymptotic cones of such 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 Behrstock and Minsky, we show that the mapping class group has this property, and recover the rank theorem of Behrstock and Minsky and of Hamenstädt. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Pacific Journal of Mathematics
Publisher:	University of California, Berkeley
Place of Publication:	Coventry
ISSN:	0030-8730
Date:	28 February 2013
Volume:	Volume 261
Number:	Number 1
Page Range:	pp. 53-93
Identification Number:	10.2140/pjm.2013.261.53
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Related URLs:	Author
URI:	http://wrap.warwick.ac.uk/id/eprint/48816

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