On equivalence relations induced by locally compact TSI Polish groups admitting open identity component
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916459262771200 |
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| author | Zheng, Yang |
| author_facet | Zheng, Yang |
| contents | For a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^ω/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$.
We prove a Rigid theorem on locally compact TSI Polish groups admitting open identity component, as follows: Let $G$ be a locally compact TSI Polish group such that $G_0$ is open in $G$, and let $H$ be a nontrivial pro-Lie TSI Polish group. Then $E(G)\leq_BE(H)$ iff there exists a continuous homomorphism $ϕ:G_0\to H_0$ satisfying the following conditions: (i) $\ker(ϕ)$ is non-archimedean; (ii) $ϕ{\rm Inn}_G(G_0)\subseteq\overline{{\rm Inn}_H(H_0)ϕ}$ under pointwise convergence topology.
An application of the Rigid theorem yields a negative answer to Question 7.5 of [2]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On equivalence relations induced by locally compact TSI Polish groups admitting open identity component Zheng, Yang Logic 03E15, 22A05, 22D05, 22E15, 46A1 For a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^ω/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. We prove a Rigid theorem on locally compact TSI Polish groups admitting open identity component, as follows: Let $G$ be a locally compact TSI Polish group such that $G_0$ is open in $G$, and let $H$ be a nontrivial pro-Lie TSI Polish group. Then $E(G)\leq_BE(H)$ iff there exists a continuous homomorphism $ϕ:G_0\to H_0$ satisfying the following conditions: (i) $\ker(ϕ)$ is non-archimedean; (ii) $ϕ{\rm Inn}_G(G_0)\subseteq\overline{{\rm Inn}_H(H_0)ϕ}$ under pointwise convergence topology. An application of the Rigid theorem yields a negative answer to Question 7.5 of [2]. |
| title | On equivalence relations induced by locally compact TSI Polish groups admitting open identity component |
| topic | Logic 03E15, 22A05, 22D05, 22E15, 46A1 |
| url | https://arxiv.org/abs/2410.21851 |