Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Abstract

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] 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 N^(n+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 π_1(N) is hyperbolic relative to π_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.