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 2 aleph 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 aleph 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 aleph 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).

Automated summary

This study uses basic tools of descriptive set theory to prove that a closed set S of marked groups has 2 aleph 0 quasi-isometry classes, and further analyzes the perfect sets of marked groups with dense subsets of finitely presented groups. The results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups, and show the existence of 2 aleph 0 quasi-isometry classes of finitely generated groups with interesting algebraic, geometric, or model-theoretic properties.