Higher rank lattices are not coarse median

TL;DR

The paper proves that higher rank lattices and certain symmetric spaces do not admit coarse median structures, implying they are not quasi-isometric to CAT(0) cube complexes, thus answering a question by Haglund.

Contribution

It establishes the non-existence of coarse median structures in higher rank symmetric spaces and lattices, providing new insights into their geometric properties.

Findings

Higher rank symmetric spaces lack coarse median structures.

Lattices in simple higher rank groups are not coarse median.

These spaces are not quasi-isometric to CAT(0) cube complexes.

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.