Virtual Critical Regularity of Mapping Class Group Actions on the Circle

We show that if G<inf>1</inf> and G<inf>2</inf> are non-solvable groups, then no C1,τ action of (G<inf>1</inf>×G<inf>2</inf>)∗ℤ on S1 is faithful for τ > 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 Mod(S) with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.