Asymptotic geometry of negatively curved manifolds of finite volume

Abstract

We study the asymptotic behaviour of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice. If the quotient manifoldX = \X is asymptotically 1/4-pinched, we prove that is divergent and UX 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 R , where is the topological entropy of the geodesic flow ofX. This generalizes Margulis' celebrated theorem to negatively curved spaces of finite volume. In contrast, we exhibit examples of lattices 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.

The weak mixing property on negatively curved manifolds

“…This measure is in fact the unique measure of maximal entropy built by Bowen and Margulis for Anosov and axiom A flows if the nonwandering set of the flow is compact. Examples by Dal'Bo et al showed that this measure may be infinite when the nonwandering set of the flow is non-compact and the curvature is variable, in which case there is no probability invariant measure maximising the entropy [DPPS19].…”