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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Igor Belegradek

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 Mn with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+1)–manifold Nn+1 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 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