On equivalence relations induced by locally compact abelian Polish groups

TL;DR

The paper investigates when the Borel equivalence relation E(G) induced by a locally compact abelian Polish group G is reducible to E(H) for another such group H. It develops a rigid theorem linking E(G) ≤_B E(H) to a continuous homomorphism S: G_0→H_0 with non-archimedean kernel, and proves a dual rigid theorem connecting E(G) ≤_B E(H) to a nonzero continuous dual map S^*: ĝ(H)→ĝ(G) with torsion in the quotient ĝ(G)/im(S^*). Finite-dimensional and P-adic solenoid phenomena are analyzed: dim(G)=rank(ĝ) for finite-dimensional compact G, E(R^n) ≤_B E(G) iff 𝕋^n embeds in G, and E(R^n) <_B E(G) ≤_B E(T^n) iff dim(G)=n; Σ_P is characterized by E(Σ_P) ≤_B E(Σ_Q) iff Q≼P, and P(ω)/Fin embeds into the Borel-reducibility order between E(ℝ^n) and E(𝕋^n). The results connect descriptive set theory with abelian topological groups, providing structural criteria and revealing both straightforward and intricate cases across dimensions.

Abstract

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 $G$. 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 $P(ω)/\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb R}^n)$ and $E({\mathbb T}^n)$.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2: Rigid Theorem
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Dual Rigid Theorem
• Definition 2.1: DZ
• Proposition 2.2
• proof
• Proposition 2.3: DZ