This talk is mainly about the surface subgroups.
The geometry of surface subgroups of the mapping class group
is essentially equivalent to the geometry of surface-by-surface groups.
There are many open questions, for example it is not known if a
surface-by-surface group can be hyperbolic, or indeed if it can
be ``atoroidal'' in the sense of not containing any free abelian
subgroup of rank 2. However one can show that, for fixed genera,
there are only finitely many isomorphism classes of atoroidal
surface-by-surface groups. The proof uses the geometry of the
curve graph of Harvey and ideas of Masur and Minsky et al. from the
proof of the ending lamination conjecture.