This article is cited in 16 papers

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

L. A. Beklaryan

Central Economics and Mathematics Institute, RAS

Abstract: In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group $G\subseteq \operatorname {Homeo}(S^1)$ of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on $S^1$. In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group $G\subseteq \operatorname {Homeo}(S^1)$ exists if and only if the quotient group $G/H_G$ does not contain a free subgroup with two generators (here $H_G$ is some specific subgroup of $G$ defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups $G\subseteq \operatorname {Homeo}(\mathbb R)$ of homeomorphisms of the line.

UDC: 515.1

Received: 29.03.2001
Revised: 29.08.2001

DOI: 10.4213/mzm350

 English version: Mathematical Notes, 2002, 71:3, 305–315

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