$C^1$ actions on the circle of finite index subgroups of $Mod({\Sigma }_g)$, $Aut(F_n)$, and $Out(F_n)$

Abstract

Let ${\Sigma }_g$ be a closed, connected, and oriented surface of genus $g\ge 24$ and let $\Gamma$ be a finite index subgroup of the mapping class group $Mod({\Sigma }_g)$ that contains the Torelli group $\mathcal{I}({\Sigma }_g)$. Then any orientation preserving $C^1$ action of $\Gamma$ on the circle cannot be faithful. We also show that if $\Gamma$ is a finite index subgroup of $Aut(F_n)$, when $n\ge 8$, that contains the subgroup of IA-automorphisms, then any orientation preserving $C^1$ action of $\Gamma$ on the circle cannot be faithful. Similarly, if $\Gamma$ is a finite index subgroup of $Out(F_n)$, when $n\ge 8$, that contains the Torelli group ${\mathcal{T}}_n$, then any orientation preserving $C^1$ action of $\Gamma$ on the circle cannot be faithful. In fact, when $n\ge 10$, any orientation preserving $C^1$ action of a finite index subgroup of $Aut(F_n)$ or $Out(F_n)$ on the circle cannot be faithful.

arXiv comment12 pages. Corrected typos. Incorporated referee comments. Paper has been accepted. arXiv admin note: text overlap with arXiv:0803.4281