Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces
| Autor: | Julien Melleray, Mahmood Etedadialiabadi, François Le Maître, Su Gao |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Finite group Group (mathematics) General Mathematics 010102 general mathematics Mathematics::General Topology 0102 computer and information sciences Group Theory (math.GR) Mathematics - Logic Urysohn and completely Hausdorff spaces Automorphism 01 natural sciences Primary 03C13 03C55 Secondary 20E06 20E26 Metric space Mathematics::Logic 010201 computation theory & mathematics Locally finite group Isometry FOS: Mathematics Mathematics::Metric Geometry 0101 mathematics Isometry group Logic (math.LO) Mathematics - Group Theory Mathematics |
| Popis: | 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 $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fra\"iss\'{e} class, where Hall's group appears as the acting group of the Fra\"iss\'{e} 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 $\Delta$-metric spaces in terms of the distance value set $\Delta$. Comment: 37 pages |
| Databáze: | OpenAIRE |
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