On equivalence relations induced by Polish groups admitting compatible two-sided invariant metrics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917911508025344 |
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| author | Ding, Longyun Zheng, Yang |
| author_facet | Ding, Longyun Zheng, Yang |
| contents | Given 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 first established two results:
(1) Let $G,H$ be two Polish groups. If $H$ is TSI but $G$ is not, then $E(G)\not\le_BE(H)$.
(2) Let $G$ be a Polish group. Then the following are equivalent: (a) $G$ is TSI non-archimedean; (b)$E(G)\leq_B E_0^ω$; and (c) $E(G)\leq_B{\mathbb R}^ω/c_0$. In particular, $E(G)\sim_B E_0^ω$ iff $G$ is TSI uncountable non-archimedean.
A critical theorem presented in this article is as follows: Let $G$ be a TSI Polish group, and let $H$ be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le_BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker(S)$ is non-archimedean, where $G_0$ is the connected component of the identity of $G$. The converse holds if $G$ is connected, $S(G)$ is closed in $H$, and the interval $[0,1]$ can be embedded into $H$.
As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On equivalence relations induced by Polish groups admitting compatible two-sided invariant metrics Ding, Longyun Zheng, Yang Logic 03E15, 22A05, 22D05, 22E15, 46A16 Given 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 first established two results: (1) Let $G,H$ be two Polish groups. If $H$ is TSI but $G$ is not, then $E(G)\not\le_BE(H)$. (2) Let $G$ be a Polish group. Then the following are equivalent: (a) $G$ is TSI non-archimedean; (b)$E(G)\leq_B E_0^ω$; and (c) $E(G)\leq_B{\mathbb R}^ω/c_0$. In particular, $E(G)\sim_B E_0^ω$ iff $G$ is TSI uncountable non-archimedean. A critical theorem presented in this article is as follows: Let $G$ be a TSI Polish group, and let $H$ be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le_BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker(S)$ is non-archimedean, where $G_0$ is the connected component of the identity of $G$. The converse holds if $G$ is connected, $S(G)$ is closed in $H$, and the interval $[0,1]$ can be embedded into $H$. As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. |
| title | On equivalence relations induced by Polish groups admitting compatible two-sided invariant metrics |
| topic | Logic 03E15, 22A05, 22D05, 22E15, 46A16 |
| url | https://arxiv.org/abs/2401.15556 |