#### 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
##### Abstract

Let $n\ge 3$, let $M$ be an orientable complete finite-volume hyperbolic $n$–manifold with compact (possibly empty) geodesic boundary, and let $Vol\left(M\right)$ and $\parallel M\parallel$ be the Riemannian volume and the simplicial volume of $M$. A celebrated result by Gromov and Thurston states that if $\partial M=\varnothing$ then $Vol\left(M\right)∕\parallel M\parallel ={v}_{n}$, 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 $\partial M\ne \varnothing$ then $Vol\left(M\right)∕\parallel M\parallel <{v}_{n}$.

We prove here that for every $\eta >0$ there exists $k>0$ (only depending on $\eta$ and $n$) such that if $Vol\left(\partial M\right)∕Vol\left(M\right)\le k$, then $Vol\left(M\right)∕\parallel M\parallel \ge {v}_{n}-\eta$. As a consequence we show that for every $\eta >0$ there exists a compact orientable hyperbolic $n$–manifold  $M$ with nonempty geodesic boundary such that $Vol\left(M\right)∕\parallel M\parallel \ge {v}_{n}-\eta$.

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
##### 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