Orbital functions and heat kernels of Kleinian groups
[Fonctions orbitales et noyaux de la chaleur des groupes kleiniens]
Boulanger, Adrien1
1 Institut Mathématique de Marseille, CNRS, Aix-Marseille Université 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France
Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1069-1100.

Nous étudions les fonctions orbitales des groupes kleiniens par l’approche du noyau de la chaleur initiée dans [Bou22].

We study orbital functions associated to Kleinian groups through the heat kernel approach developed in [Bou22].

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.200
Classification : 11F72, 30F40, 51M10, 58J35
Keywords: Kleinian groups, heat kernels, orbital functions
Mot clés : Groupes kleiniens, noyaux de la chaleur, fonctions orbitales
Boulanger, Adrien 1

1 Institut Mathématique de Marseille, CNRS, Aix-Marseille Université 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France 
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Boulanger, Adrien. Orbital functions and heat kernels of Kleinian groups. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1069-1100. doi : 10.5802/jep.200. http://archive.numdam.org/articles/10.5802/jep.200/

[BCM12] Brock, Jeffrey F.; Canary, Richard D.; Minsky, Yair N. The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2), Volume 176 (2012) no. 1, pp. 1-149 | DOI | MR | Zbl

[BFZ02] Babillot, Martine; Feres, Renato; Zeghib, Abdelghani Rigidité, groupe fondamental et dynamique, Panoramas & Synthèses, 13, Société Mathématique de France, Paris, 2002

[BJ97a] Bishop, Christopher J.; Jones, Peter W. Hausdorff dimension and Kleinian groups, Acta Math., Volume 179 (1997) no. 1, pp. 1-39 | DOI | MR | Zbl

[BJ97b] Bishop, Christopher J.; Jones, Peter W. The law of the iterated logarithm for Kleinian groups, Lipa’s legacy (New York, 1995) (Contemp. Math.), Volume 211, American Mathematical Society, Providence, RI, 1997, pp. 17-50 | DOI | MR | Zbl

[Bou18] Boulanger, Adrien Quelques exemples de systèmes dynamiques: comptage en mesure infinie, enlacement sur le tore et échanges d’intervalles affines, Ph. D. Thesis, Sorbonne Université (2018) | TEL

[Bou22] Boulanger, Adrien A stochastic approach to counting problems, Ann. Sci. École Norm. Sup. (4), Volume 55 (2022) no. 1, pp. 225-259 | DOI | MR | Zbl

[Can93] Canary, Richard D. Ends of hyperbolic 3-manifolds, J. Amer. Math. Soc., Volume 6 (1993) no. 1, pp. 1-35 | DOI | MR | Zbl

[Can08] Canary, Richard D. Marden’s tameness conjecture: history and applications, Geometry, analysis and topology of discrete groups (Adv. Lect. Math. (ALM)), Volume 6, Int. Press, Somerville, MA, 2008, pp. 137-162 | MR | Zbl

[CG06] Calegari, Danny; Gabai, David Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc., Volume 19 (2006) no. 2, pp. 385-446 | DOI | MR | Zbl

[Cha84] Chavel, Isaac Eigenvalues in Riemannian geometry, Pure and applied math., 115, Academic Press, 1984

[Del42] Delsarte, Jean Sur le gitter fuchsien, C. R. Acad. Sci. Paris, Volume 214 (1942), pp. 147-179 | MR | Zbl

[Dod83] Dodziuk, Jozef Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., Volume 32 (1983) no. 5, pp. 703-716 | DOI | MR | Zbl

[EM93] Eskin, Alex; McMullen, Curt Mixing, counting, and equidistribution in Lie groups, Duke Math. J., Volume 71 (1993) no. 1, pp. 181-209 | DOI | MR | Zbl

[Eps85] Epstein, Charles L. The spectral theory of geometrically periodic hyperbolic 3-manifolds, Mem. Amer. Math. Soc., 58, no. 335, American Mathematical Society, Providence, RI, 1985 | DOI | MR

[Eps87] Epstein, Charles L. Asymptotics for closed geodesics in a homology class, the finite volume case, Duke Math. J., Volume 55 (1987) no. 4, pp. 717-757 | DOI | MR | Zbl

[Eps89] Epstein, Charles L. Positive harmonic functions on abelian covers, J. Funct. Anal., Volume 82 (1989) no. 2, pp. 303-315 | DOI | MR | Zbl

[GN98] Grigor’yan, Alexander; Noguchi, Masakazu The heat kernel on hyperbolic space, Bull. London Math. Soc., Volume 30 (1998) no. 6, pp. 643-650 | DOI | MR | Zbl

[Gri09] Grigor’yan, Alexander Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Math., 47, American Mathematical Society, Providence, RI, 2009

[Hub56] Huber, Heinz Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I, Comment. Math. Helv., Volume 30 (1956), pp. 20-62 | DOI | Zbl

[LP82] Lax, Peter D.; Phillips, Ralph S. The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal., Volume 46 (1982) no. 3, pp. 280-350 | DOI | MR | Zbl

[LST81] Geometry Symposium, Utrecht 1980, Lect. Notes in Math., 894 (1981) | DOI

[Mar69] Margulis, G. A. Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen., Volume 3 (1969) no. 4, pp. 89-90 | MR

[McM96] McMullen, Curtis T. Renormalization and 3-manifolds which fiber over the circle, Annals of Math. Studies, 142, Princeton University Press, Princeton, NJ, 1996 | DOI

[Min02] Minsky, Yair N. End Invariants and the classification of hyperbolic 3-manifolds, Current Developments in Math. (2002), pp. 111-141 https://projecteuclid.org:443/euclid.cdm/1088530401

[Ota01] Otal, Jean-Pierre The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, 7, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001 Translated from the 1996 French original (Astérisque, vol. 235) | MR

[Pat75] Patterson, S. J. A lattice-point problem in hyperbolic space, Mathematika, Volume 22 (1975) no. 1, pp. 81-88 | DOI | MR | Zbl

[Pat76] Patterson, S. J. The limit set of a Fuchsian group, Acta Math., Volume 136 (1976) no. 3-4, pp. 241-273 | DOI | MR | Zbl

[Pat88] Patterson, S. J. On a lattice-point problem in hyperbolic space and related questions in spectral theory, Ark. Mat., Volume 26 (1988) no. 1, pp. 167-172 | DOI | MR | Zbl

[Rob02] Roblin, Thomas Sur la fonction orbitale des groupes discrets en courbure négative, Ann. Inst. Fourier (Grenoble), Volume 52 (2002) no. 1, pp. 145-151 | DOI | Numdam | MR | Zbl

[Rob03] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. France (N.S.), 95, Société Mathématique de France, Paris, 2003 | Numdam | MR

[Sel56] Selberg, Atle Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), Volume 20 (1956), pp. 47-87 | MR | Zbl

[Sel60] Selberg, Atle On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp. 147-164 | Zbl

[Sul79] Sullivan, Dennis The density at infinity of a discrete group of hyperbolic motions, Publ. Math. Inst. Hautes Études Sci., Volume 50 (1979), pp. 171-202 | DOI | Numdam | Zbl

[Sul84] Sullivan, Dennis Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., Volume 153 (1984) no. 3-4, pp. 259-277 | DOI | MR | Zbl

[Sul87] Sullivan, Dennis Related aspects of positivity in Riemannian geometry, J. Differential Geom., Volume 25 (1987) no. 3, pp. 327-351 http://projecteuclid.org/euclid.jdg/1214440979 | MR | Zbl

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