The simplicial volume of hyperbolic manifolds with geodesic boundary

Frigerio, Roberto; Pagliantini, Cristina

doi:10.2140/agt.2010.10.979

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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

DOI: 10.2140/agt.2010.10.979

Abstract

Let $n \geq 3$ , 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 $\partial M = \emptyset$ 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 $\partial M \neq \emptyset$ 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 $Vol (\partial M) ∕ Vol (M) \leq k$ , then $Vol (M) ∕ ∥ M ∥ \geq v_{n} - η$ . 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 $Vol (M) ∕ ∥ M ∥ \geq v_{n} - η$ .

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 2000

Primary: 53C23

Secondary: 57N16, 57N65

References

Publication

Received: 7 November 2009

Revised: 14 March 2010

Accepted: 18 March 2010

Published: 23 April 2010

Authors

Roberto Frigerio
Dipartimento di Matematica “L Tonelli” Università di Pisa Largo B Pontecorvo 5 I-56127 Pisa Italy
http://www.dm.unipi.it/~frigerio/
Cristina Pagliantini
Dipartimento di Matematica “L Tonelli” Università di Pisa Largo B Pontecorvo 5 I-56127 Pisa Italy

Volume 10, issue 2 (2010)