#### Volume 6, issue 3 (2006)

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Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Algebraic & Geometric Topology 6 (2006) 1341–1354
 arXiv: math.GR/0509504
##### 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.

##### Keywords
hyperbolic, relatively hyperbolic, hyperbolization of polyhedra, aspherical manifold, simplicial volume, assembly map, Novikov Conjecture
Primary: 20F65
##### Publication
Received: 19 October 2005
Accepted: 30 June 2006
Published: 20 September 2006
##### Authors
 Igor Belegradek School of Mathematics Georgia Institute of Technology Atlanta GA 30332–0160