Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

Belegradek, Igor¹

¹School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332–0160

Algebraic and Geometric Topology, Tome 6 (2006) no. 3, pp. 1341-1354

Résumé

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.

Détail

DOI : 10.2140/agt.2006.6.1341

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

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