Quasi-isometric diversity of marked groups

We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2ℵ0 quasi-isometry classes provided 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 2ℵ0 quasiisometry 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 2ℵ0 quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties (e.g., such groups can be torsion, simple, verbally complete or they can all have the same elementary theory).