Simplicial volume and fillings of hyperbolic manifolds

Fujiwara, Koji¹ ; Manning, Jason²

¹Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan
²Department of Mathematics, University at Buffalo, SUNY, Buffalo NY 14260, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 4, pp. 2237-2264

Résumé

Let M be a hyperbolic n–manifold whose cusps have torus cross-sections. In an earlier paper, the authors constructed a variety of nonpositively and negatively curved spaces as “2π–fillings” of M by replacing the cusps of M with compact “partial cones” of their boundaries. These 2π–fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2π–filling is positive, and bounded above by Vol(M) vn , where vn is the volume of a regular ideal hyperbolic n–simplex. This result generalizes the fact that hyperbolic Dehn filling of a 3–manifold does not increase hyperbolic volume.

In particular, we obtain information about the simplicial volumes of some 4–dimensional homology spheres described by Ratcliffe and Tschantz, answering a question of Belegradek and establishing the existence of 4–dimensional homology spheres with positive simplicial volume.

Détail

DOI : 10.2140/agt.2011.11.2237

Classification : 20F65, 53C23
Keywords: simplicial volume, pseudomanifold, Dehn filling

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