Quasi-isometric diversity of marked groups

TL;DR

This paper employs descriptive set theory to demonstrate that certain closed sets of marked groups contain continuum many quasi-isometry classes, especially when every open subset includes non-quasi-isometric groups, highlighting the abundance of diverse group structures.

Contribution

It establishes that closed sets with dense finitely presented groups contain continuum many quasi-isometry classes, advancing understanding of the diversity of finitely generated groups.

Findings

Most known constructions of non-quasi-isometric groups are encompassed.

Any perfect set with a dense finitely presented subset has continuum many classes.

The results enable the existence of many groups with specific algebraic or geometric properties.

Abstract

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ 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. They can also be used to prove the existence of $2^{\aleph_0}$ quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties.