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 [Bel] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+ 1)manifold N with hyperbolic fundamental group. (II) If B1, . . . Bm 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 Bk , and π1(N) is hyperbolic relative to π1(Bk) ’s. (III) Any finite aspherical simplicial complex is a retract of a closed aspherical triangulated manifold with positive simplicial volume and nonelementary relatively hyperbolic fundamental group.