C1 actions on the circle of finite index subgroups of Mod(Σg), Aut(Fn), and Out(Fn)

Abstract

Let Σg be a closed, connected, and oriented surface of genus g≥24, and let Γ be a finite index subgroup of the mapping class group Mod(Σg) that contains the Torelli group ℐ(Σg). Then any orientation-preserving C1 action of Γ on the circle cannot be faithful. We also show that if Γ is a finite index subgroup of Aut(Fn), when n≥8, that contains the subgroup of IA-automorphisms, then any orientation-preserving C1 action of Γ on the circle cannot be faithful. Similarly, if Γ is a finite index subgroup of Out(Fn), when n≥8, that contains the Torelli group 𝒯n, then any orientation preserving C1 action of Γ on the circle cannot be faithful. In fact, when n≥10, any orientation-preserving C1 action of a finite index subgroup of Aut(Fn) or Out(Fn) on the circle cannot be faithful.