Actions of Non-Compact and Non-Locally Compact Polish Groups

Journal of Symbolic Logic 65 (4):1881-1894 (2000)
  Copy   BIBTEX


We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.



    Upload a copy of this work     Papers currently archived: 92,813

External links

  • This entry has no external links. Add one.
Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Actions of non-compact and non-locally compact polish groups.Sławomir Solecki - 2000 - Journal of Symbolic Logic 65 (4):1881-1894.
Turbulence Phenomena in Real Analysis.Nikolaos Efstathiou Sofronidis - 2005 - Archive for Mathematical Logic 44 (7):801-815.
Calculus of variations and descriptive set theory.Nikolaos E. Sofronidis - 2009 - Mathematical Logic Quarterly 55 (5):535-538.
Quasi-Polish spaces.Matthew de Brecht - 2013 - Annals of Pure and Applied Logic 164 (3):356-381.
Analytic equivalence relations and bi-embeddability.Sy-David Friedman & Luca Motto Ros - 2011 - Journal of Symbolic Logic 76 (1):243 - 266.
Diagonal Actions and Borel Equivalence Relations.Longyun Ding & Su Gao - 2006 - Journal of Symbolic Logic 71 (4):1081 - 1096.
Polish group actions and effectivity.Barbara Majcher-Iwanow - 2012 - Archive for Mathematical Logic 51 (5-6):563-573.
Open subspaces of locally compact metric spaces.Mark Mandelkern - 1993 - Mathematical Logic Quarterly 39 (1):213-216.
G-compactness and groups.Jakub Gismatullin & Ludomir Newelski - 2008 - Archive for Mathematical Logic 47 (5):479-501.
Universally measurable subgroups of countable index.Christian Rosendal - 2010 - Journal of Symbolic Logic 75 (3):1081-1086.
Constructive compact operators on a Hilbert space.Hajime Ishihara - 1991 - Annals of Pure and Applied Logic 52 (1-2):31-37.


Added to PP


6 months

Historical graph of downloads

Sorry, there are not enough data points to plot this chart.
How can I increase my downloads?

Citations of this work

Countable borel equivalence relations.S. Jackson, A. S. Kechris & A. Louveau - 2002 - Journal of Mathematical Logic 2 (01):1-80.

Add more citations

References found in this work

No references found.

Add more references