论文标题
关于波兰群体引起的等效关系
On equivalence relations induced by Polish groups
论文作者
论文摘要
本文的动机是介绍一种轨道等效关系，可以从降低鲍尔（Borel）的角度很好地描述波兰群体的结构和特性。给定波兰组$ g $，让$ e（g）$是正确的coset等价关系$ g^ω/c（g）$，其中$ c（g）$是$ g $中所有收敛序列的组。 让$ g $成为波兰团体。 （1）$ g $是一个离散可计可数组，包含至少两个元素，如果f $ e（g）\ sim_be_0 $; （2）如果$ g $是tsi不可数的非架构，则$ e（g）\ sim_be_0^ω$; （3）$ g $是非架构的iff $ e（g）\ le_b =^+$; （4）如果$ h $是CLI抛光群，但$ g $不是，则$ e（g）\ not \ le_be（h）$; （5）如果$ h $是一个非Archimedean波兰人组，但$ g $不是，则$ e（g）\ not \ le_be（h）$。 引入了$α$ -L.M.-l.m.-不平衡的$α<ω_1$的概念。令$ g，h $为波兰群，$ 0 <α<ω_1$。如果$ g $是$α$ -L.M.-M.-M.-nboranced，但$ h $不是，则$ e（g）\ not \ le_b e（h）$。 对于TSI抛光群，将减少孔的存在转化为拓扑组之间良好行为的连续映射的存在。作为其应用程序，对于任何波兰组$ g $，令$ g_0 $是身份元素$ 1_g $的连接组件。令$ g $和$ h $为两个可分离的TSI谎言组。如果$ e（g）\ le_be（h）$，则存在连续的本地注入映射$ s：g_0 \ to H_0 $。此外，如果$ g_0，H_0 $是Abelian，则$ S $是组同构。特别是，对于$ C_0，E_0，C_1，E_1 \ in {\ Mathbb n} $，$ e（{\ Mathbb r}^{C_0} \ times {\ times {\ Mathbb t}^{e_0}^{e_0}） t}^{e_1}）$ iff $ e_0 \ le e_1 $和$ c_0+e_0 \ le c_1+e_1 $。
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 $c_0+e_0\le c_1+e_1$.
