Volume 11, issue 4 (2011)

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Simplicial volume and fillings of hyperbolic manifolds

Koji Fujiwara and Jason Fox Manning

Algebraic & Geometric Topology 11 (2011) 2237–2264

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.

simplicial volume, pseudomanifold, Dehn filling
Mathematical Subject Classification 2010
Primary: 20F65, 53C23
Received: 2 February 2011
Accepted: 3 June 2011
Published: 25 August 2011
Koji Fujiwara
Graduate School of Information Sciences
Tohoku University
Jason Fox Manning
Department of Mathematics
University at Buffalo, SUNY
Buffalo NY 14260