Algebraic & Geometric Topology 10 (2010) 979–1001

DOI: 10.2140/agt.2010.10.979

Abstract

Let n ≥ 3, let M be an orientable complete finite-volume hyperbolic n–manifold with compact (possibly empty) geodesic boundary, and let Vol(M) and ∥M∥ be the Riemannian volume and the simplicial volume of M. A celebrated result by Gromov and Thurston states that if ∂M = ∅ then Vol(M) ∕ ∥M∥ = v_n, where v_n is the volume of the regular ideal geodesic n–simplex in hyperbolic n–space. On the contrary, Jungreis and Kuessner proved that if ∂M≠∅ then Vol(M) ∕ ∥M∥ < v_n.

We prove here that for every η > 0 there exists k > 0 (only depending on η and n) such that if Vol(∂M) ∕ Vol(M) ≤ k, then Vol(M) ∕ ∥M∥≥ v_n − η. As a consequence we show that for every η > 0 there exists a compact orientable hyperbolic n–manifold  M with nonempty geodesic boundary such that Vol(M) ∕ ∥M∥≥ v_n − η.

Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for noncompact finite-volume hyperbolic n–manifolds without geodesic boundary.

Keywords

Gromov norm, straight simplex, hyperbolic volume, Haar measure, volume form

Mathematical Subject Classification

Primary: 53C23

Secondary: 57N16, 57N65

References

Publication

Received: 7 November 2009
Revised: 14 March 2010
Accepted: 18 March 2010
Published: 23 April 2010

Authors