Asymptotic geometry of negatively curved manifolds of finite volume

We study the asymptotic behavior of simply connected Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice Γ. If the quotient manifold X = Γ\X is asymptotically 1=4-pinched, we prove that Γ is divergent and U X has finite Bowen-Margulis measure (which is then ergodic and totally conservative with respect to the geodesic flow); moreover, we show that, in this case, the volume growth of balls B(x,R) in X is asymptotically equivalent to a purely exponential function c.x/eδR, where δ is the topological entropy of the geodesic flow of X . This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices Γ in negatively curved spaces X (not asymptotically 1/4-pinched) where, depending on the critical exponent of the parabolic subgroups and on the finiteness of the Bowen- Margulis measure, the growth function is exponential, lower-exponential or even upper-exponential.