On equivalence relations induced by Polish groups

TL;DR

This paper introduces a new class of orbit equivalence relations for Polish groups, analyzing their complexity via Borel reducibility and establishing connections with topological group mappings.

Contribution

It defines and studies the properties of orbit equivalence relations induced by Polish groups, including their classification and relation to Borel reducibility, with applications to Lie groups and topological mappings.

Findings

Characterization of $E(G)$ for discrete countable groups

Classification of $E(G)$ for TSI uncountable non-archimedean groups

Conditions for Borel reducibility between orbit equivalence relations

Abstract

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 $G^\omega/c(G)$, 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 $E(G)\sim_BE_0^\omega$; (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 $\alpha$-l.m.-unbalanced Polish group for $\alpha<\omega_1$ is introduced. Let $G,H$ be Polish groups, $0<\alpha<\omega_1$. If $G$ is $\alpha$-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 $c_0+e_0\le c_1+e_1$.