$C^1$ actions on the circle of finite index subgroups of $Mod(Σ_g)$, $Aut(F_n)$, and $Out(F_n)$

Abstract: Let $\Sigma_{g}$ be a closed, connected, and oriented surface of genus $g \geq 24$ 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 $n \geq 8$, 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 $n \geq 8$, 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 $n \geq 10$, 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.