On equivalence relations induced by Polish groups admitting compatible two-sided invariant metrics

TL;DR

The work analyzes the descriptive-set-theoretic complexity of equivalence relations $E(G)$ arising from Polish groups, linking Borel reducibility to the algebraic-topological structure of the groups. It introduces a Pre-rigid Theorem for TSI groups and derives a broad family of rigid theorems across Lie, locally compact, pro-Lie, Banach, and Fréchet spaces, clarifying when reductions must come from continuous homomorphisms with non-archimedean kernels. A key finding is the precise characterization: a Polish group is TSI non-archimedean iff $E(G)$ is reducible to $E_0^ω$ and to ${b R}^ω/c_0$, and there is a sharp gap between $E_0^ω$ and ${b R}^ω/c_0$ with no intermediate $G$. The results culminate in strong rigidity phenomena, including local isomorphisms under uniform NSS assumptions, and concrete corollaries for Lie groups, pro-Lie groups, Banach spaces, and Fréchet spaces, illustrating how descriptive-set-theoretic complexity encodes deep structural properties.

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 $G$. We first established two results: (1) Let $G,H$ be two Polish groups. If $H$ is TSI but $G$ is not, then $E(G)\not\le_BE(H)$. (2) Let $G$ be a Polish group. Then the following are equivalent: (a) $G$ is TSI non-archimedean; (b)$E(G)\leq_B E_0^ω$; and (c) $E(G)\leq_B{\mathbb R}^ω/c_0$. In particular, $E(G)\sim_B E_0^ω$ iff $G$ is TSI uncountable non-archimedean. A critical theorem presented in this article is as follows: Let $G$ be a TSI Polish group, and let $H$ be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le_BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker(S)$ is non-archimedean, where $G_0$ is the connected component of the identity of $G$. The converse holds if $G$ is connected, $S(G)$ is closed in $H$, and the interval $[0,1]$ can be embedded into $H$. As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively.

Theorems & Definitions (77)

• Theorem 1.1: Rigid Theorem, DZlcoabe
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: Rigid Theorem for locally compact TSI groups
• Theorem 1.8: Rigid Theorem for TSI Lie groups
• Theorem 1.9: Rigid Theorem for Fréchet spaces
• Theorem 1.10: Rigid Theorem for Banach spaces