Algebraic & Geometric Topology

Higher rank lattices are not coarse median

Thomas Haettel

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2895-2910.

Received: 23 June 2015
Revised: 5 January 2016
Accepted: 6 February 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 51E24: Buildings and the geometry of diagrams 51F99: None of the above, but in this section 53C35: Symmetric spaces [See also 32M15, 57T15]

median algebra coarse geometry quasi-isometry higher rank lattice symmetric space building CAT (0) cube complex


Haettel, Thomas. Higher rank lattices are not coarse median. Algebr. Geom. Topol. 16 (2016), no. 5, 2895--2910. doi:10.2140/agt.2016.16.2895.

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