Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [3] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M-n with hyperbolic fundamental group is a retract of a closed aspherical triangulated. (n+1) -manifold Nn+1 with hyperbolic fundamental group. (II) If B-1, ... B-m are closed aspherical triangulated n-manifolds, then there is a closed aspherical triangulated manifold N of dimension n+1 such that N has nonzero simplicial volume, N retracts to each B-k, and pi(1)(N) is hyperbolic relative to pi(1)(B-k)'s. (III) Any finite aspherical simplicial complex is a retract of a closed aspherical triangulated manifold with positive simplicial volume and non-elementary relatively hyperbolic fundamental group.