2015/03/13 by Françoise Dal’Bo, Dal'Bo, F., Marc Peigné +5
Computer Science · Mathematics · #37C35 #53C20 #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #F.0 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1503.03971
openalex publication_date 2015/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the asymptotic behaviour of simply connected, Riemannian manifolds X of strictly negative curvature admitting a non-uniform lattice Γ. If the quotient manifold X= Γ\backslash 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. \linebreak 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.