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2010 The simplicial volume of hyperbolic manifolds with geodesic boundary
Roberto Frigerio, Cristina Pagliantini
Algebr. Geom. Topol. 10(2): 979-1001 (2010). DOI: 10.2140/agt.2010.10.979


Let n3, 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)Mvnη. 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)Mvnη.

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.


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Roberto Frigerio. Cristina Pagliantini. "The simplicial volume of hyperbolic manifolds with geodesic boundary." Algebr. Geom. Topol. 10 (2) 979 - 1001, 2010.


Received: 7 November 2009; Revised: 14 March 2010; Accepted: 18 March 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1206.53045
MathSciNet: MR2629772
Digital Object Identifier: 10.2140/agt.2010.10.979

Primary: 53C23
Secondary: 57N16 , 57N65

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

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 2 • 2010
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