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.
Highlights
Quasi-isometry is an equivalence relation that identifies metric spaces having the same large scale geometry. It is especially useful in geometric group theory and plays essentially the same role as the isomorphism relation in algebra
It is well-known that there exist 2א0 quasi-isometry classes of finitely generated groups
A simpler argument using small cancellation theory was given by Bowditch in [Bow]
Summary
Quasi-isometry is an equivalence relation that identifies metric spaces having the same large scale geometry. There exist 2א0 pairwise non-quasi-isometric finitely generated groups which simultaneously satisfy all properties listed in Proposition 1.3. The same approach can be used to construct uncountable sets of pairwise non-quasi-isometric finitely generated groups with other interesting algebraic properties, e.g., torsion-free Tarski Monsters and groups with 2 conjugacy classes. We briefly discuss these applications at the end of Section 4. The proof is a simple combination of Corollary 1.2 and a recent result of the second author [Osi[21], Theorem 2.9] stating that H0 contains a comeagre elementary equivalence class. Note that there is no hope to prove the same result for solvable groups of derived length 2 as finitely generated metabelian groups satisfy the maximum condition for normal subgroups [Hal] and, the set of such groups is countable
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