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

TL;DR

This paper characterizes when certain equivalence relations from locally compact TSI Polish groups are reducible to those from pro-Lie TSI Polish groups, revealing structural conditions and providing a negative answer to a prior open question.

Contribution

It establishes a rigid criterion for Borel reducibility between equivalence relations induced by specific classes of Polish groups, advancing understanding of their structural relationships.

Findings

Characterizes Borel reducibility via continuous homomorphisms with specific kernel properties.

Provides a negative answer to an open question in the field.

Identifies conditions under which equivalence relations are comparable.

Abstract

For 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 $G$. 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 $\phi:G_0\to H_0$ satisfying the following conditions: (i) $\ker(\phi)$ is non-archimedean; (ii) $\phi{\rm Inn}_G(G_0)\subseteq\overline{{\rm Inn}_H(H_0)\phi}$ under pointwise convergence topology. An application of the Rigid theorem yields a negative answer to Question 7.5 of [2].