Asymptotic geometry of negatively curved manifolds of finite volume

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $\mathrm{\Gamma }$. If the quotient manifold $\bar{X}=\mathrm{\Gamma }\mathrm{\backslash }X$ is asymptotically $1\mathrm{/}4$-pinched, we prove that $\mathrm{\Gamma }$ is divergent and $U\bar{X}$ 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 $c(x)e^{\delta R}$, where $\delta$ is the topological entropy of the geodesic flow of $\bar{X}$. \linebreak This generalizes Margulis’ celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices $\mathrm{\Gamma }$ in negatively curved spaces $X$ (not asymptotically $1\mathrm{/}4$-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.