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Orbital functions and heat kernels of Kleinian groups
[Fonctions orbitales et noyaux de la chaleur des groupes kleiniens]
Adrien Boulanger1
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
Adrien Boulanger 1

1 Institut Mathématique de Marseille, CNRS, Aix-Marseille Université 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Adrien Boulanger. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.200/

[BCM12] J. F. Brock, R. D. Canary & Y. N. Minsky - “The classification of Kleinian surface groups, II: The ending lamination conjecture”, Ann. of Math. (2) 176 (2012) no. 1, p. 1-149 | DOI | MR | Zbl

[BFZ02] M. Babillot, R. Feres & A. Zeghib - Rigidité, groupe fondamental et dynamique, Panoramas & Synthèses, vol. 13, Société Mathématique de France, Paris, 2002

[BJ97a] C. J. Bishop & P. W. Jones - “Hausdorff dimension and Kleinian groups”, Acta Math. 179 (1997) no. 1, p. 1-39 | DOI | MR | Zbl

[BJ97b] C. J. Bishop & P. W. Jones - “The law of the iterated logarithm for Kleinian groups”, in Lipa’s legacy (New York, 1995), Contemp. Math., vol. 211, American Mathematical Society, Providence, RI, 1997, p. 17-50 | DOI | MR | Zbl

[Bou18] A. Boulanger - 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] A. Boulanger - “A stochastic approach to counting problems”, Ann. Sci. École Norm. Sup. (4) 55 (2022) no. 1, p. 225-259 | DOI | MR | Zbl

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

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

[CG06] D. Calegari & D. Gabai - “Shrinkwrapping and the taming of hyperbolic 3-manifolds”, J. Amer. Math. Soc. 19 (2006) no. 2, p. 385-446 | DOI | MR | Zbl

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

[Del42] J. Delsarte - “Sur le gitter fuchsien”, C. R. Acad. Sci. Paris 214 (1942), p. 147-179 | MR | Zbl

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

[EM93] A. Eskin & C. McMullen - “Mixing, counting, and equidistribution in Lie groups”, Duke Math. J. 71 (1993) no. 1, p. 181-209 | DOI | MR | Zbl

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

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

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

[GN98] A. Grigor’yan & M. Noguchi - “The heat kernel on hyperbolic space”, Bull. London Math. Soc. 30 (1998) no. 6, p. 643-650 | DOI | MR | Zbl

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

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

[LP82] P. D. Lax & R. S. Phillips - “The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces”, J. Funct. Anal. 46 (1982) no. 3, p. 280-350 | DOI | MR | Zbl

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

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

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

[Min02] Y. N. Minsky - “End Invariants and the classification of hyperbolic 3-manifolds”, Current Developments in Math. (2002), p. 111-141

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

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

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

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

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

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

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

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

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

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

[Sul87] D. Sullivan - “Related aspects of positivity in Riemannian geometry”, J. Differential Geom. 25 (1987) no. 3, p. 327-351 | MR | Zbl

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