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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Classification of the group actions on the real line and circle


Author: A. V. Malyutin
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 2.
Journal: St. Petersburg Math. J. 19 (2008), 279-296
MSC (2000): Primary 54H15; Secondary 57S25, 57M60, 54H20, 37E05, 37E10
Published electronically: February 7, 2008
MathSciNet review: 2333902
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Abstract | References | Similar Articles | Additional Information

Abstract: The group actions on the real line and circle are classified. It is proved that each minimal continuous action of a group on the circle is either a conjugate of an isometric action, or a finite cover of a proximal action. It is also shown that each minimal continuous action of a group on the real line either is conjugate to an isometric action, or is a proximal action, or is a cover of a proximal action on the circle. As a corollary, it is proved that a continuous action of a group on the circle either has a finite orbit, or is semiconjugate to a minimal action on the circle that is either isometric or proximal. As a consequence, a new proof of the Ghys-Margulis alternative is obtained.


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Additional Information

A. V. Malyutin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: malyutin@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-08-00999-0
PII: S 1061-0022(08)00999-0
Keywords: Circle, line, group of homeomorphisms, action, proximal, distal, semiconjugacy
Received by editor(s): June 16, 2006
Published electronically: February 7, 2008
Additional Notes: The author was partially supported by grant NSh-4329.2006.1 and by RFBR grant no. 05-01-00899
Article copyright: © Copyright 2008 American Mathematical Society