Higher rank lattices are not coarse median

Abstract

We show that symmetric spaces and thick affine buildings which are not of spherical type A1 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.