Volume 16, issue 5 (2016)

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Higher rank lattices are not coarse median

Thomas Haettel

Algebraic & Geometric Topology 16 (2016) 2895–2910
Abstract

We show that symmetric spaces and thick affine buildings which are not of spherical type A1r have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.

Keywords
median algebra, coarse geometry, quasi-isometry, higher rank lattice, symmetric space, building, CAT (0) cube complex
Mathematical Subject Classification 2010
Primary: 20F65, 51E24, 51F99, 53C35
References
Publication
Received: 23 June 2015
Revised: 5 January 2016
Accepted: 6 February 2016
Published: 7 November 2016
Authors
Thomas Haettel
Université de Montpellier
Institut Montpelliérain Alexander Grothendieck
CC051
Place Eugène Bataillon
34095 Montpellier Cedex 5
France
http://www.math.univ-montp2.fr/~haettel/