Coarse median property of virtually nilpotent groups

Abstract

We show that virtually nilpotent groups are coarse median if and only if they are virtually abelian. The main idea is that the sub-Riemannian geometry of the asymptotic cone obstructs the existence of a locally convex Lipschitz median of finite rank. As an application, we deduce that non-compact lattices in the isometry group of a rank 1 symmetric space of non-compact type other than real hyperbolic space are not coarse median. This establishes the remaining case in the classification of lattices with the coarse median property initiated by Haettel. The same approach applies more generally to complete finite-volume non-compact Riemannian manifolds $M$ of pinched negative sectional curvature: if at least one cusp cross-section does not admit a flat metric, then $\pi_1(M)$ is not coarse median.