Non-rectifiable Delone sets in SOL and other solvable groups

TL;DR

This paper constructs examples of coarsely dense subsets in certain solvable groups, such as SOL and Baumslag-Solitar groups, that are not biLipschitz equivalent to the original lattices, revealing new geometric properties.

Contribution

It demonstrates the existence of non-rectifiable, coarsely dense subsets in lattices of SOL and related solvable groups, expanding understanding of their geometric structures.

Findings

Existence of coarsely dense subsets not biLipschitz equivalent to lattices in SOL

Similar results established for higher rank abelian-by-abelian groups

Results extended to solvable Baumslag-Solitar groups

Abstract

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.