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

Dal’Bo, Françoise; Peigné, Marc

Ann. Inst. Fourier 46, No. 3, 755-799 (1996).

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

Keywords:

parabolic isometry; cuspidal point; transfer operator; harmonic renewal theorem

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