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ON EQUIVALENCE RELATIONS INDUCED BY POLISH GROUPS ADMITTING COMPATIBLE TWO-SIDED INVARIANT METRICS

Part of: Lie groups Topological linear spaces and related structures Locally compact groups and their algebras Topological and differentiable algebraic systems Set theory

Published online by Cambridge University Press:  03 February 2025

LONGYUN DING Open the ORCID record for LONGYUN DING [Opens in a new window]  and
YANG ZHENG
LONGYUN DING
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN, 300071 P.R. CHINA E-mail: dingly@nankai.edu.cn
YANG ZHENG*
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES AND LPMC NANKAI UNIVERSITY TIANJIN, 300071 P.R. CHINA E-mail: dingly@nankai.edu.cn
*
e-mail: 1120200015@mail.nankai.edu.cn
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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. We first established two results:

  1. (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$.

  2. (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^\omega $; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In particular, $E(G)\sim _B E_0^\omega $ 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.

Keywords

Borel reductionTSI Polish groupequivalence relation

MSC classification

Primary: 03E15: Descriptive set theory 22A05: Structure of general topological groups
Secondary: 22D05: General properties and structure of locally compact groups 22E15: General properties and structure of real Lie groups 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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