Asymptotic geometry of negatively curved manifolds of finite volumeReportar como inadecuado

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1 IRMAR - Institut de Recherche Mathématique de Rennes 2 LMPT - Laboratoire de Mathématiques et Physique Théorique 3 Università degli Studi di Roma -La Sapienza- Rome

Abstract : 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\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 Bx, R in X is asymptotically equivalent to a purely exponential function cxe δR , 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.

Keywords : 37C35 Keywords: Cartan-Hadamard manifold AMS classification : 53C20 volume entropy Bowen-Margulis measure

Autor: Françoise Dal-Bo - Marc Peigné - Jean-Claude Picaud - Andrea Sambusetti -



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