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

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 $\mathbb{Z}∕2\mathbb{Z}$.