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Free subgroups of the homeomorphism group of the circle. (English. Abridged French version) Zbl 0983.37029

From the author’s abstract: “Tits’ alternative says that a finitely generated linear group either contains a noncommutative free subgroup or is virtually solvable. It is known that an analogue of Tits’ alternative is not true for subgroups of the group Homeo\((S^1)\) of all homeomorphisms of the circle \(S^1\) and even for subgroups of the group of \(C^\infty\)-diffeomorphisms of \(S^1\). The main purpose of this Note is to prove a conjecture of E. Ghys which can be viewed as a replacement of Tits’ alternative for Homeo\((S^1)\) and which says that if \(G\) is a subgroup of Homeo\((S^1)\) containing no free noncommutative subgroup then there is a \(G\)-invariant probability measure on \(S^1\).”
The paper under review is well written and organized and has high level.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
57S25 Groups acting on specific manifolds
37E10 Dynamical systems involving maps of the circle
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