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 [?] 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 $\left(n+1\right)$–manifold $N^{n+1}$ with hyperbolic fundamental group.

(II) If $B_1,\dots B_m$ 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 ${\pi }_1\left(N\right)$ is hyperbolic relative to ${\pi }_1\left(B_k\right)$’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.