For every positive integer n, we exhibit a cofinite subgroup Gn of the mapping class group of a surface of genus at most two such that Gn admits an epimorphism onto a free group of rank n. We conclude that H1 (Gn; {\mathbb Z}) has rank at least n and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum. In the case of genus two, the groups Gn can be chosen not to contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of a subgroup of finite index in the mapping class group of a sphere with four punctures (or a torus) such that it is not the restriction of an endomorphism of the whole group.

For every positive integer n, we exhibit a cofinite subgroup Gn of the mapping class group of a surface of genus at most two such that Gn admits an epimorphism onto a free group of rank n. We conclude that H1 (Gn; {\mathbb Z}) has rank at least n and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum. In the case of genus two, the groups Gn can be chosen not to contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of a subgroup of finite index in the mapping class group of a sphere with four punctures (or a torus) such that it is not the restriction of an endomorphism of the whole group.