Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds (Q1127908)

For a Haken 3-manifold \(M\) with incompressible boundary, it is proven 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, it is shown that if \(F\) is a nonempty compact 2-manifold in \(\partial M\) such that \(\partial M-F\) is incompressible, then the classifying space \(\text{BDiff}(M \text{rel} F)\) of the diffeomorphism group of \(M\) relative to \(F\) has the homotopy type of a finite aspherical complex.