# Unsmoothable group actions on compact one-manifolds

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

Background
1. 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.
2. (Farb--Franks) All Torelli groups and braid groups virtually admit faithful C1 actions on the circle.
3. (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
1. The group < x, y, z, w | [x,y] = [y,z] = [z,w] = 1 > does not admit a faithful C2 action on the circle.
2. (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.
3. 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.
4. The n-strand braid group virtually admits a faithful C2 action on the circle iff n is at most three.