Geodesic
On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line
Matematičeskie zametki, Tome 71 (2002) no. 3, pp. 334-347.

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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. 
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L. A. Beklaryan. On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line. Matematičeskie zametki, Tome 71 (2002) no. 3, pp. 334-347. http://geodesic.mathdoc.fr/item/MZM_2002_71_3_a1/

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