On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line |
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Authors: | L A Beklaryan |
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Institution: | (1) Central Institute of Mathematical Economics, Russian Academy of Sciences, Russia |
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Abstract: | In 1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group
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
exists if and only if the quotient group
does not contain a free subgroup with two generators (here
is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups
of homeomorphisms of the line. |
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Keywords: | Tits alternative homeomorphism group of the circle homeomorphism group of the line invariant probabilistic measure |
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