# Unsmoothable group actions on compact one-manifolds

From w

Written by Hyungryul Baik, Sang-hyun Kim and Thomas Koberda.

- Background

- 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 C
^{1}actions on the circle. - (Rufus Bowen's Notebook, attributed to Sullivan--Thurston; possible reinterpretation) Is the Nielsen's action of Mod
_{g}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 C
^{2}action on the circle. - (Question by Farb) Let S be a surface. Then Mod(S) virtually admits a faithful C
^{2}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 C
^{2}action on the circle iff the genus of S is at most two. - The n-strand braid group virtually admits a faithful C
^{2}action on the circle iff n is at most three.