The simplicial volume of hyperbolic manifolds with geodesic boundary

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.