4.2 Article

Quasi-isometric diversity of marked groups

期刊

JOURNAL OF TOPOLOGY
卷 14, 期 2, 页码 488-503

出版社

WILEY
DOI: 10.1112/topo.12187

关键词

20F69; 20F65 (primary); 03E15; 03C60 (secondary)

资金

  1. NSF [DMS-1612473]
  2. Feodor Lynen Research Fellowship of the Humboldt Foundation
  3. DFG Heisenberg grant [WI 4079/6]

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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.
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).

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