On equivalence relations induced by locally compact TSI Polish groups admitting open identity component

Abstract

For 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 prove a Rigid theorem on locally compact TSI Polish groups admitting open identity component, as follows: Let $G$ be a locally compact TSI Polish group such that $G_0$ is open in $G$, and let $H$ be a nontrivial pro-Lie TSI Polish group. Then $E(G)\leq_BE(H)$ iff there exists a continuous homomorphism $φ:G_0\to H_0$ satisfying the following conditions: (i) $\ker(φ)$ is non-archimedean; (ii) $φ{\rm Inn}_G(G_0)\subseteq\overline{{\rm Inn}_H(H_0)φ}$ under pointwise convergence topology. An application of the Rigid theorem yields a negative answer to Question 7.5 of [2].

Theorems & Definitions (35)

• Theorem 1.1: Rigid theorem on locally compact TSI Polish groups admitting open identity component
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1: DZtsi