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

Abstract

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group $G \subseteq {\text{Homeo}}(S^1 )$ 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 $G \subseteq {\text{Homeo}}(S^1 )$ 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 $G \subseteq {\text{Homeo}}(\mathbb{R})$ of homeomorphisms of the line.