Unsmoothable group actions on compact one-manifolds
- The complexity of a surface is \(3g-3+p+b\) where \((g,p,b)\) are the genus, the number of punctures and the number of boundary components.
- (Farb--Franks) All Torelli groups and braid groups virtually admit faithful C1 actions on the circle.
- (Rufus Bowen's Notebook, attributed to Sullivan--Thurston; possible reinterpretation) Is the Nielsen's action of Modg conjugate to a smooth action on the circle?
- Main Theorem
- The group < x, y, z, w | [x,y] = [y,z] = [z,w] = 1 > does not admit a faithful C2 action on the circle.
- (Question by Farb) Let S be a surface. Then Mod(S) virtually admits a faithful C2 action on the circle iff the complexity of S is at most one.
- Let S be a closed surface. Then the Torelli group of S virtually admits a faithful C2 action on the circle iff the genus of S is at most two.
- The n-strand braid group virtually admits a faithful C2 action on the circle iff n is at most three.