  We prove a bound relating the volume of a curve near a cusp in a hyperbolic
manifold to its multiplicity at the cusp. The proof uses a hybrid technique
employing both the geometry of the uniformizing group and the algebraic
geometry of the toroidal compactification. There are a number of consequences:
we show that for an $n$-dimensional toroidal compactification $\bar X$ with
boundary $D$, $K_{\bar X}+(1-\frac{n+1}{2\pi}) D$ is nef, and in particular
that $K_{\bar X}$ is ample for $n\geq 6$. By an independent algebraic argument,
we prove that every hyperbolic manifold of dimension $n\geq 3$ is of general
type, and conclude that the phenomena famously exhibited by Hirzebruch in
dimension 2 do not occur in higher dimensions. Finally, we investigate the
applications to the problem of bounding the number of cusps and to the
Green--Griffiths conjecture.
