  We prove that the mapping class group of a surface obtained from removing a
Cantor set from either the 2-sphere, the plane, or the interior of the closed
2-disk has no proper countable-index subgroups. The proof is an application of
the automatic continuity of these groups, which was established by Mann. As
corollaries, we see that these groups do not contain any proper finite-index
subgroups and that each of these groups have trivial abelianization.
