Hostname: page-component-7b9c58cd5d-sk4tg Total loading time: 0 Render date: 2025-03-13T11:13:09.377Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Becker, H., Polish group actions: Dichotomies and generalized elementary embeddings . Journal of the American Mathematical Society , vol. 11 (1998), pp. 397–449.Google Scholar
Becker, H. and Kechris, A. S., The Descriptive Set Theory of Polish Group Actions , London Mathematical Society Lecture Note Series, 232, Cambridge University Press, Cambridge, 1996.Google Scholar
Ding, L., Borel reducibility and Hölder $\left(\alpha \right)$ embeddability between Banach spaces . The Journal of Symbolic Logic , vol. 77 (2012), pp. 224–244.Google Scholar
Ding, L., On equivalence relations generated by Schauder bases . The Journal of Symbolic Logic , vol. 82 (2017), pp. 1459–1481.Google Scholar
Ding, L. and Zheng, Y., On equivalence relations induced by Polish groups, preprint, 2022, arXiv:2204.04594, Unpublished.Google Scholar
Ding, L. and Zheng, Y., On equivalence relations induced by locally compact abelian Polish groups . The Journal of Symbolic Logic , (2023), pp. 1–16. https://doi.org/10.1017/jsl.2023.35.Google Scholar
Farah, I., Basis problem for turbulent action II: ${c}_0$ -equalities . Proceedings of the London Mathematical Society , vol. 82 (2001), pp. 1–30.Google Scholar
Gao, S., Invariant Descriptive Set Theory , Monographs and Textbooks in Pure and Applied Mathematics, 293, CRC Press, Boca Raton, 2009.Google Scholar
Hofmann, K. H. and Morris, S. A., The Lie Theory of Connected Pro-Lie Groups , EMS Tracts in Mathematics, 2, European Mathematical Society, Zürich, 2007.Google Scholar
Hofmann, K. H. and Morris, S. A., The Structure of Compact Groups , De Gruyter Studies in Mathematics, 25, De Gruyter, Berlin, 2013.Google Scholar
Hofmann, K. H., Morris, S. A., and Stroppel, M., Varieties of topological groups, Lie groups, and SIN-groups . Colloqoquim Mathematicum , vol. 70 (1996), pp. 151–163.Google Scholar
Hofmann, K. H., Morris, S. A., and Stroppel, M., Locally compact groups, residual Lie groups, and varieties generated by Lie group . Topology and its Applications , vol. 71 (1996), pp. 63–91.Google Scholar
Kechris, A. S., Classical Descriptive Set Theory , Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.Google Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I: Sequence Spaces , Ergebnisse der Mathematik und ihrer Grenzgebiete, 92, Springer-Verlag, Berlin, 1977.Google Scholar
Morris, S. A. and Pestov, V., On Lie groups in varieties of topological groups . Colloquim Mathematicae , vol. 78 (1998), pp. 39–47.Google Scholar
Rosendal, C., Lipschitz structure and minimal metrics on topological groups . Arkiv for Matematik , vol. 56 (2018), pp. 185–206.Google Scholar
Stiles, W. J., On properties of subspaces of ${l}_p$ , $0<p<1$ . Transactions of the American Mathematical Society , vol. 149 (1970), pp. 405–415.Google Scholar
Upmeier, H., Symmetric Banach Manifolds and Jordan ${\boldsymbol{C}}^{\ast}$ -Algebras , North-Holland Mathematics Studies, 104, North-Holland, Amsterdam, 1985.Google Scholar
Varadarajan, V. S., Lie Groups, Lie Algebras, and their Representations , Graduate Texts in Mathematics, 102, Springer-Verlag, New York, 1974.Google Scholar
Wilansky, A., Modern Methods in Topological Vector Spaces , Mcgraw-Hill International Book Company, New York, 1978.Google Scholar
Wojtaszczyk, P., Banach Spaces for Analysts , Cambridge Studies in Advanced Mathematics, 25, Cambridge University Press, Cambridge, 1991.Google Scholar