  We prove a homological stability theorem for the subgroup of the mapping
class group acting as the identity on some fixed portion of the first homology
group of the surface. We also prove a similar theorem for the subgroup of the
mapping class group preserving a fixed map from the fundamental group to a
finite group, which can be viewed as a mapping class group version of a theorem
of Ellenberg-Venkatesh-Westerland about braid groups. These results require
studying various simplicial complexes formed by subsurfaces of the surface,
generalizing work of Hatcher-Vogtmann.
