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

Abstract

Let Σ g be a closed, connected, and oriented surface of genus g ≥ 24 and let Γ be a finite index subgroup of the mapping class group M od(Σ g ) that contains the Torelli group I(Σ g ). Then any orientation preserving C 1 action of Γ on the circle cannot be faithful.We also show that if Γ is a finite index subgroup of Aut(F n ), when n ≥ 8, that contains the subgroup of IA-automorphisms, then any orientation preserving C 1 action of Γ on the circle cannot be faithful.Similarly, if Γ is a finite index subgroup of Out(F n ), when n ≥ 8, that contains the Torelli group T n , then any orientation preserving C 1 action of Γ on the circle cannot be faithful.In fact, when n ≥ 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.