A group G is called ‘good’ if the canonical map $G\to\hat{G}$ to the profinite completion induces isomorphisms $H^i(\hat{G},M)\to H^i(G,M)$ for any finite $G$-module $M$. I’ve had multiple academics in my department claim to me that the orientation-preserving mapping class group of an orientable surface of genus $g$ with $n$ boundary components satisfies this property, though none have been able to give me a reference.

I think the easiest way to prove this would be to exhibit a finite index solvable subgroup, which for mapping class groups is equivalent to a finite index abelian subgroup (A theorem in Sullivan’s ‘genetics of homotopy theory’ says this implies that G is good). I was thinking of the subgroup of homeomorphisms which act trivially on cohomology with $\mathbb{Z}/n\mathbb{Z}$ coefficients, but I have very little experience with solvable groups and couldn’t even figure out if this was a reasonable guess.

So, does anyone know if there exists a finite index solvable subgroup of the mapping class groups? Or is there a different proof that the mapping class groups are good?

Thanks for any help.