Simplicial volume and fillings of hyperbolic manifolds

Let $M$ be a hyperbolic $n$-manifold whose cusps have torus
cross-sections. We constructed a variety of nonpositively and  negatively
curved spaces as "$2\pi$-fillings" of $M$ by replacing the  cusps of $M$
with compact "partial cones" of their boundaries.  We  show that the
simplicial volume of any such $2\pi$-filling is  positive, and bounded
above by Vol$(M)/v_n$, where $v_n$ is the volume  of a regular ideal
hyperbolic $n$-simplex. This result generalizes the  fact that hyperbolic
Dehn filling of a 3-manifold does not increase  hyperbolic volume. This is
a joint work with J. Manning.