Dense Locally Finite Subgroups of Automorphism Groups of Ultraextensive Spaces

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Article verifying a conjecture of Vershik by showing that Hall’s universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. It shows the same for all automorphism groups of known infinite ultraextensive spaces.

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37 p.

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Etedadialiabadi, Mahmood; Gao, Su; Le Maître, François & Melleray, Julien August 18, 2021.

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Article verifying a conjecture of Vershik by showing that Hall’s universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. It shows the same for all automorphism groups of known infinite ultraextensive spaces.

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37 p.

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Abstract: We verify a conjecture of Vershik by showing that Hall’s universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal Kₙ-free graph for all n ≥ 3. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a FraÏssé class, where Hall’s group appears as the acting group of the FraÏssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn ∆-metric spaces in terms of the distance value set ∆.

This is the preprint version of the article, reprinted with permission from Elsevier Science Ltd., all rights reserved. The final definitive version is available here: https://doi.org/10.1016/j.aim.2021.107966

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  • Advances in Mathematics, 391(107966), Elsevier Science Ltd., August 18 2021, pp. 1-37

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  • Publication Title: Advances in Mathematics
  • Volume: 391
  • Article Identifier: 107966
  • Peer Reviewed: Yes

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  • August 18, 2021

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  • Sept. 22, 2021, 2:32 p.m.

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  • Nov. 14, 2023, 1:28 p.m.

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Etedadialiabadi, Mahmood; Gao, Su; Le Maître, François & Melleray, Julien. Dense Locally Finite Subgroups of Automorphism Groups of Ultraextensive Spaces, article, August 18, 2021; (https://digital.library.unt.edu/ark:/67531/metadc1838860/: accessed April 20, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Science.

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