Volume 10, issue 2 (2010)

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The simplicial volume of hyperbolic manifolds with geodesic boundary

Roberto Frigerio and Cristina Pagliantini

Algebraic & Geometric Topology 10 (2010) 979–1001

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.

Gromov norm, straight simplex, hyperbolic volume, Haar measure, volume form
Mathematical Subject Classification 2000
Primary: 53C23
Secondary: 57N16, 57N65
Received: 7 November 2009
Revised: 14 March 2010
Accepted: 18 March 2010
Published: 23 April 2010
Roberto Frigerio
Dipartimento di Matematica “L Tonelli”
Università di Pisa
Largo B Pontecorvo 5
I-56127 Pisa
Cristina Pagliantini
Dipartimento di Matematica “L Tonelli”
Università di Pisa
Largo B Pontecorvo 5
I-56127 Pisa