Virtual Critical Regularity of Mapping Class Group Actions on the Circle

We show that if G1 and G2 are non-solvable groups, then no C1,τ action of $(G_{1}\times G_{2})*\mathbb {Z}$ on S1 is faithful for τ > 0. As a corollary, S an orientable surface complexity at least three the critical regularity arbitrary finite index subgroup mapping class group Mod(S) with respect to circle most one, thus strengthening result first two authors Baik.