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\(C^1\) actions on the mapping class groups on the circle. (English) Zbl 1155.37028

Summary: Let \(S\) be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any \(C^{1}\) action of the mapping class group of \(S\) on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have \(C^{1}\) faithful actions on the circle. We also prove that for n \(\geq 6\), any \(C^{1}\) action of \(\operatorname{Aut}(F_{n})\) or \(\text{Out}(F_{n})\) on the circle factors through an action of \(\mathbb Z/2\mathbb Z\).

MSC:

37E10 Dynamical systems involving maps of the circle
57M60 Group actions on manifolds and cell complexes in low dimensions
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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