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 that 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. 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).

中文翻译：

---

标记组的拟等距多样性

我们使用描述性集合论的基本工具来证明一个封闭集合 $\mathcal{小号}$ 的标记组有 ${\mathrm{2个}}^{{\aleph }_0}$ 准等距类，条件是 $\mathcal{小号}$至少包含两个非拟等距组。因此，每组具有有限表示的组的密集子集的标记组的每一个完美集合都包含${\mathrm{2个}}^{{\aleph }_0}$准等距类。这些结果说明了非拟等距有限生成群的连续族的大多数已知构造。我们用它们来证明存在${\mathrm{2个}}^{{\aleph }_0}$ 具有有趣的代数，几何或模型理论特性的有限生成组的拟等距类（例如，此类组可以是扭转，简单，口头完整的，也可以具有相同的基本理论）。