Asymptotic geometry of negatively curved manifolds of finite volume

TL;DR

This paper extends Margulis' theorem to negatively curved manifolds of finite volume, analyzing their asymptotic geometry, volume growth, and ergodic properties under specific curvature pinching conditions.

Contribution

It proves that asymptotically 1/4-pinched negatively curved manifolds have divergent lattices with finite Bowen-Margulis measure, generalizing Margulis' theorem to finite volume spaces.

Findings

Volume growth is asymptotically exponential with rate δ.

Manifolds with 1/4-pinched curvature have ergodic geodesic flow.

Examples show different growth behaviors depending on curvature pinching and measure finiteness.

Abstract

We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $\Gamma$. If the quotient manifold $\bar X= \Gamma \backslash X$ is asymptotically $1/4$-pinched, we prove that $\Gamma$ 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 $\Gamma$ 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.