Non rectifiable delone sets in SOL and other solvable groups

Abstract

Given a lattice Gamma subset of SOL, we show there is a uniformly discrete coarsely dense subset D subset of Gamma that is not bi-Lipschitz equivalent to Gamma. We also prove similar results for lattices in certain higher rank abelian-by-abelian groups and for the solvable Baumslag-Solitar groups.