#### 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 ${A}_{1}^{r}$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT$\left(0\right)$ 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