Non-rectifiable Delone sets in SOL and other solvable groups

Given a lattice $\Gamma \subset SOL$, we show that there is a coarsely dense subset $\mathcal{D} \subset \Gamma$ that is not biLipschitz 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.