Let , let be an orientable complete finite-volume hyperbolic –manifold with compact (possibly empty) geodesic boundary, and let and be the Riemannian volume and the simplicial volume of . A celebrated result by Gromov and Thurston states that if then , where is the volume of the regular ideal geodesic –simplex in hyperbolic –space. On the contrary, Jungreis and Kuessner proved that if then .
We prove here that for every there exists (only depending on and ) such that if , then . As a consequence we show that for every there exists a compact orientable hyperbolic –manifold with nonempty geodesic boundary such that .
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 –manifolds without geodesic boundary.
"The simplicial volume of hyperbolic manifolds with geodesic boundary." Algebr. Geom. Topol. 10 (2) 979 - 1001, 2010. https://doi.org/10.2140/agt.2010.10.979