Higher rank lattices are not coarse median

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(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.

13 pages, 2 figures. To appear in Algebraic & Geometric Topology

Cited by in corpus (4)

• Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
• A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups
• Superrigidity of actions on finite rank median spaces
• CAT(0) Polygonal Complexes are 2-Median