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.
Keywords
Gromov norm, straight simplex, hyperbolic volume, Haar
measure, volume form