Simplicial volume and fillings of hyperbolic manifolds

Abstract

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 “$2\pi$–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 $\frac{Vol\left(M\right)}{v_n}$, 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.