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arXiv 2204.04594math.LO
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由波兰群诱导等价关系

On equivalence relations induced by Polish groups

School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院及LPMC）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）

Longyun Ding, Yang Zheng

备注：
45 pages, submitted
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AI总结：

本文研究了由波兰群诱导的等价关系，探讨了其在Borel可约性中的性质及应用，提出了一系列关于波兰群的等价关系判定条件。

摘要：

本文的动机是引入一种轨道等价关系，可以从Borel可约性角度很好地描述波兰群的结构和性质。给定一个波兰群G，令E(G)为右陪集等价关系G^ω/c(G)，其中c(G)是G中所有收敛序列的群。设G为波兰群。(1)如果G是一个离散可数群且至少有两个元素，则E(G)~B E_0；(2)如果G是TSI不可数非阿基米德群，则E(G)~B E_0^ω；(3)如果G是非阿基米德群，则E(G)≤B=^+；(4)如果H是CLI波兰群但G不是，则E(G)≠≤B E(H)；(5)如果H是非阿基米德波兰群但G不是，则E(G)≠≤B E(H)。引入了α-l.m.-unbalanced波兰群的概念，其中α<ω₁。设G,H为波兰群，0<α<ω₁。如果G是α-l.m.-unbalanced但H不是，则E(G)≠≤B E(H)。对于TSI波兰群，Borel可约性的存在性转化为拓扑群之间存在良好行为的连续映射的存在性。作为应用，对于任何波兰群G，令G₀为单位元1_G的连通分量。设G和H为两个可分的TSI李群。如果E(G)≤B E(H)，则存在连续局部单射映射S:G₀→H₀。此外，如果G₀,H₀是阿贝尔群，则S是群同态。特别地，对于c₀,e₀,c₁,e₁∈N，E(R^{c₀}×T^{e₀})≤B E(R^{c₁}×T^{e₁})当且仅当e₀≤e₁且c₀+e₀≤c₁+e₁。

英文摘要：

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. 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 $G$. Let $G$ be a Polish group. (1) $G$ is a discrete countable group containing at least two elements iff $E(G)\sim_BE_0$; (2) if $G$ is TSI uncountable non-archimedean, then $E(G)\sim_BE_0^ω$; (3) $G$ is non-archimedean iff $E(G)\le_B=^+$; (4) if $H$ is a CLI Polish group but $G$ is not, then $E(G)\not\le_BE(H)$; (5) if $H$ is a non-archimedean Polish group but $G$ is not, then $E(G)\not\le_BE(H)$. The notion of $α$-l.m.-unbalanced Polish group for $α<ω_1$ is introduced. Let $G,H$ be Polish groups, $0<α<ω_1$. If $G$ is $α$-l.m.-unbalanced but $H$ is not, then $E(G)\not\le_B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group $G$, let $G_0$ be the connected component of the identity element $1_G$. Let $G$ and $H$ be two separable TSI Lie groups. If $E(G)\le_BE(H)$, then there exists a continuous locally injective map $S:G_0\to H_0$. Moreover, if $G_0,H_0$ are abelian, $S$ is a group homomorphism. In particular, for $c_0,e_0,c_1,e_1\in{\mathbb N}$, $E({\mathbb R}^{c_0}\times{\mathbb T}^{e_0})\le_BE({\mathbb R}^{c_1}\times{\mathbb T}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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