Quasi‐isometric diversity of marked groups. Issue 2 (5th April 2021) - Minasyan, A.; Osin, D.; Witzel, S.
Abstract: We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2 ℵ 0 quasi‐isometry classes, provided that every non‐empty open subset of S contains at least two non‐quasi‐isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2 ℵ 0 quasi‐isometry classes. These results account for most known constructions of continuous families of non‐quasi‐isometric finitely generated groups. We use them to prove the existence of 2 ℵ 0 quasi‐isometry classes of finitely generated groups having interesting algebraic, geometric, or model‐theoretic properties (for example, such groups can be torsion, simple, verbally complete or they can all have the same elementary theory).