Simplicial volume and fillings of hyperbolic manifolds

Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In [ 8], 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.