Volume 8, issue 2 (2008)

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$C^1$ actions on the mapping class groups on the circle

Kamlesh Parwani

Algebraic & Geometric Topology 8 (2008) 935–944
Abstract

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 $n\ge 6$, any ${C}^{1}$ action of $Aut\left({F}_{n}\right)$ or $Out\left({F}_{n}\right)$ on the circle factors through an action of $ℤ∕2ℤ$.

Keywords
mapping class groups, Kazhdan groups, actions on the circle
Primary: 37E10
Secondary: 57M60