Simplicial volume and fillings of hyperbolic manifolds

TL;DR

This paper proves that certain hyperbolic manifold fillings have positive simplicial volume, bounded by the original volume, extending known results from 3-dimensional hyperbolic Dehn fillings to higher dimensions.

Contribution

It establishes positivity and bounds for the simplicial volume of 2π-fillings of hyperbolic manifolds, generalizing hyperbolic Dehn filling volume results to higher dimensions.

Findings

Simplicial volume of 2π-fillings is positive.

Bounded above by Vol(M)/v_n, where v_n is the ideal hyperbolic n-simplex volume.

Results apply to 4-dimensional homology spheres, confirming positive simplicial volume.

Abstract

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.