  Let M be a surface (possibly nonorientable) with punctures and/or boundary
components. The paper is a study of ``geometric subgroups'' of the mapping
class group of M, that is subgroups corresponding to inclusions of subsurfaces
(possibly disconnected). We characterise the subsurfaces which lead to
virtually abelian geometric subgroups. We provide algebraic and geometric
conditions under which two geometric subgroups are commensurable. We also
describe the commensurator of a geometric subgroup in terms of the stabiliser
of the underlying subsurface. Finally, we show some applications of our
analysis to the theory of irreducible unitary representations of mapping class
groups.
