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This article is cited in 14 scientific papers (total in 14 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.
Received: 29.03.2001 Revised: 29.08.2001
Citation:
L. A. Beklaryan, “On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line”, Mat. Zametki, 71:3 (2002), 334–347; Math. Notes, 71:3 (2002), 305–315
Linking options:
https://www.mathnet.ru/eng/mzm350https://doi.org/10.4213/mzm350 https://www.mathnet.ru/eng/mzm/v71/i3/p334
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