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تسجيل مستخدم جديدQuasi-isometric diversity of marked groups
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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.
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