Virtual critical regularity of mapping class group actions on the circle

TL;DR

The paper establishes that certain complex group actions on the circle cannot be smooth beyond a specific regularity, and determines the maximum regularity for mapping class groups acting on the circle.

Contribution

It proves non-faithfulness of $C^{1, au}$ actions for certain product groups and computes the critical regularity of mapping class groups on the circle.

Findings

No faithful $C^{1, au}$ actions for certain non-solvable groups on $S^1$

Critical regularity of mapping class groups on $S^1$ is at most one

Strengthens previous results on group actions on the circle

Abstract

We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,\tau}$ action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $\tau>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $\mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.