C^1 actions of the mapping class group on the circle

TL;DR

This paper proves that for surfaces with genus at least 6, the mapping class group cannot act non-trivially on the circle in a C^1 manner, and extends these results to certain automorphism groups and lattices.

Contribution

It establishes the triviality of C^1 actions of high-genus surface mapping class groups on the circle and shows similar restrictions for automorphism groups of free groups and certain lattices.

Findings

C^1 actions of high-genus mapping class groups on the circle are trivial.

Products of Kazhdan groups and certain lattices cannot have faithful C^1 actions on the circle.

Actions of Aut(F_n) and Out(F_n) for n > 5 factor through Z/2Z.

Abstract

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.