×

Ping-pong groups and closed geodesics in constant negative curvature. (Groupes du ping-pong et géodésiques fermées en courbure \(-1\).) (French) Zbl 0853.53032

Summary: We consider a class of free and discrete groups of isometries of the hyperbolic ball \({\mathbb B}^d\) which contain parabolic transformations and we prove that the number of closed geodesics on \({\mathbb B}^d/\Gamma\) whose length is less than \(a\) is equivalent to \({e^{a\delta}\over a\delta}\), where \(\delta\) is the critical exponent of the Poincaré series.

MSC:

53C22 Geodesics in global differential geometry
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] [1] , Möbius transformations in several dimensions, School of Mathematics, University of Minnesota (1981). · Zbl 0517.30001
[2] [2] , The exponent of convergence of Poincaré series, Proc. London Math. Soc., (3) 18 (1968), 461-483. · Zbl 0162.38801
[3] [3] , Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, 470. · Zbl 0308.28010
[4] [4] , & , Méthode des opérateurs de transfert : transformations dilatantes de l’intervalle et géodésiques fermées, à paraître à Astérisque.
[5] [5] & , Mixing, counting and equidistribution in Lie groups, Duke Math. Journal, 71, n° 1 (1993). · Zbl 0798.11025
[6] [6] , Fonction Zeta de Selberg et surfaces de géométrie finie, Adv. Studies in Pure Math., 21 (1992), 33-70. · Zbl 0794.58044
[7] [7] & , Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. I.H.P., n° 1 (1988), 73-98. · Zbl 0649.60041
[8] [8] & , Asymptotic winding of the geodesic flow on modular surfaces and continued fractions, Ann. Sc. E.N.S., 4ème série, t. 26 (1993), 23-50. · Zbl 0784.60076
[9] [9] , Free groups in linear groups, L’Enseignement Mathématique, 29 (1983), 129-144. · Zbl 0517.20024
[10] [10] , The selberg trace formula and the Riemann zeta function, Duke Math. J., 43 (1976), 441-482. · Zbl 0346.10010
[11] [11] , Sur un théorème spectral et son application aux noyaux Lipschitziens, Proceeding of the A.M.S., n° 118 (1993), 627-634. · Zbl 0772.60049
[12] [12] , Renewal theorems in symbolic dynamics with applications to geodesic flows, non euclidean tesselations and their fractal limits, Acta. Math., 163 (1989), 1-55. · Zbl 0701.58021
[13] [13] , Applications of ergodic theory to the investigation of manifolds of negative curvature, Funct. Anal. Appl., 3 (1969), 335-336. · Zbl 0207.20305
[14] [14] , Ergodic theory of discrete groups, Cambridge University Press, 1989. · Zbl 0674.58001
[15] [15] & , An analogue of prime number theorem for closed orbits of axiom A flows, Ann. of Prob., 118 (1983), 573-591. · Zbl 0537.58038
[16] [16] , The limit set of a Fuchsian group, Acta. Math., 136 (1976), 241-273. · Zbl 0336.30005
[17] [17] , Foundations of hyperbolic manifolds, Springer-Verlag, 1994. · Zbl 0809.51001
[18] [19] , Homotopy equivalence of 3-manifolds with boundaries, Lect. Notes Math. vol 761, 1979 · Zbl 0401.28016
[19] [19] , The density at infinity of a discrete group of hyperbolic motions, Publ. Math. IHES, vol. 50 (1979), 171-202. · Zbl 0439.30034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.