We show that if G1 and G2 are non-solvable groups, then no C1,τ action of (G1×G2)∗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.
