We show that symmetric spaces and thick affine buildings which are not of spherical type have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT 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.
"Higher rank lattices are not coarse median." Algebr. Geom. Topol. 16 (5) 2895 - 2910, 2016. https://doi.org/10.2140/agt.2016.16.2895