Virtually nilpotent groups are coarse median if and only if virtually abelian, via sub-Riemannian obstruction in the asymptotic cone, completing a lattice classification.
Higher rank lattices are not coarse median , volume=
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Coarse median property of virtually nilpotent groups
Virtually nilpotent groups are coarse median if and only if virtually abelian, via sub-Riemannian obstruction in the asymptotic cone, completing a lattice classification.