Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Belegradek, Igor

doi:10.2140/agt.2006.6.1341

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Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Igor Belegradek

Algebraic & Geometric Topology 6 (2006) 1341–1354

DOI: 10.2140/agt.2006.6.1341

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 $(n + 1)$ –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 $π_{1} (N)$ 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.

Keywords

hyperbolic, relatively hyperbolic, hyperbolization of polyhedra, aspherical manifold, simplicial volume, assembly map, Novikov Conjecture

Mathematical Subject Classification 2000

Primary: 20F65

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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

Volume 6, issue 3 (2006)