The simplicial volume of hyperbolic manifolds with geodesic boundary

Frigerio, Roberto¹ ; Pagliantini, Cristina¹

¹Dipartimento di Matematica “L Tonelli”, Università di Pisa, Largo B Pontecorvo 5, I-56127 Pisa, Italy

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 979-1001

Résumé

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∥ = vn, where vn 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∥ < vn.

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∥≥ vn − 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∥≥ vn − 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.

Détail

DOI : 10.2140/agt.2010.10.979

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

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