The simplicial volume of hyperbolic manifolds with geodesic boundary

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 parallel to M parallel to be the Riemannian volume and the simplicial volume of M. A celebrated result by Gromov and Thurston states that if partial derivative M = empty set then Vol(M)/parallel to M parallel to = upsilon(n), where upsilon(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 derivative M = empty set Vol(M)/parallel to M parallel to = upsilon(n). We prove here that for every eta > 0 there exists k > 0 ( only depending on eta and n) such that if such that if Vol(partial derivative M)/Vol(M) <= k, then Vol(M)/parallel to M parallel to >= upsilon(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(M)/parallel to M parallel to >= upsilon(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.