On equivalence relations induced by locally compact abelian Polish groups

TL;DR

This paper explores the structure of equivalence relations induced by locally compact abelian Polish groups, revealing conditions for reducibility and embedding properties related to their connected components and specific group classes.

Contribution

It establishes a characterization of Borel reducibility between these equivalence relations via continuous homomorphisms and embeddings of certain posets into the Borel equivalence relations.

Findings

If E(G) ≤_B E(H), then a continuous homomorphism from G_0 to H_0 with non-archimedean kernel exists.

The converse holds when G is connected and compact.

The poset P(ω)/Fin embeds into Borel equivalence relations between E(ℝ^n) and E(𝕋^n).

Abstract

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 $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq_B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected and compact. For $n\in{\mathbb N}^+$, the partially ordered set $P(\omega)/\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb R}^n)$ and $E({\mathbb T}^n)$.