Unsmoothable group actions on compact one-manifolds

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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.

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