Simplicial volume and fillings of hyperbolic manifolds

Details Simplicial volume and fillings of hyperbolic manifolds Simplicial volume and fillings of hyperbolic manifolds

2010-12-05

arxiv.org/abs/1012.1039v2

Mathematics

Geometric Topology

22 pages; version 2 points out the application to the homology spheres of Ratcliffe and Tschantz, and makes some other small c

Scientific paper

Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In arXiv:0901.0056, the authors constructed a variety of nonpositively and negatively curved spaces as "2\pi-fillings" of M by replacing the cusps of M with compact "partial cones" of their boundaries. These 2\pi-fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2\pi-filling is positive, and bounded above by Vol(M)/v_n, where v_n 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.

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