Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds

For a Haken 3-manifold M with incompressible boundary, we prove that the mapping class group H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in H(M) exist is given. All results are given for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in ∂M such that ∂M - F is incompressible, then the classifying space BDiff(M rel F) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.