Classification of the group actions on the real line and circle

Malyutin, A.

doi:10.1090/S1061-0022-08-00999-0

Classification of the group actions on the real line and circle
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by A. V. Malyutin
Translated by: the author

St. Petersburg Math. J. 19 (2008), 279-296

DOI: https://doi.org/10.1090/S1061-0022-08-00999-0

Published electronically: February 7, 2008

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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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