2020/04/29 by Mahmood Etedadialiabadi, Etedadialiabadi, Mahmood, Gao, Su +4 · 3 citations
Mathematics · #03C55 #20E26 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Primary 03C13 #Secondary 20E06
paper · pdf · doi:10.48550/arxiv.2004.14138
openalex publication_date 2020/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We verify a conjecture of Vershik by showing that Hall's universal countable\nlocally finite group can be embedded as a dense subgroup in the isometry group\nof the Urysohn space and in the automorphism group of the random graph. In\nfact, we show the same for all automorphism groups of known infinite\nultraextensive spaces. These include, in addition, the isometry group of the\nrational Urysohn space, the isometry group of the ultrametric Urysohn spaces,\nand the automorphism group of the universal Kn-free graph for all n\≥ 3.\nFurthermore, we show that finite group actions on finite metric spaces or\nfinite relational structures form a Fra "iss 'e class, where Hall's group\nappears as the acting group of the Fra "iss 'e limit. We also embed continuum\nmany non-isomorphic countable universal locally finite groups into the isometry\ngroups of various Urysohn spaces, and show that all dense countable subgroups\nof these groups are mixed identity free (MIF). Finally, we give a\ncharacterization of the isomorphism type of the isometry group of the Urysohn\n\Δ-metric spaces in terms of the distance value set \Δ.\n