Rigidity, counting and equidistribution of quaternionic Cartan chains
[Rigidité, comptage et équidistribution de chaînes de Cartan quaternioniennes]
Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 1, pp. 45-69.

Dans ce papier, nous montrons un analogue d’un théorème de Cartan, disant que les transformations du bord des espaces hyperboliques quaternioniens qui préservent les chaînes sont des transformations projectives. Nous donnons un résultat de comptage et d’équidistribution pour les orbites de chaînes arithmétiques dans le groupe de Heisenberg quaternionien.

In this paper, we prove an analog of Cartan’s theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

Publié le :
DOI : 10.5802/ambp.399
Classification : 11E39, 11F06, 11N45, 20G20, 53C17, 53C55
Keywords: counting, equidistribution, Cartan chain, quaternionic Heisenberg group, Cygan distance, sub-Riemannian geometry, quaternionic hyperbolic geometry
Mot clés : comptage, équidistribution, chaîne de Cartan, groupe de Heisenberg quaternionien, distance de Cygan, géométrie sous-riemanienne, géométrie hyperbolique quaternionienne
Parkkonen, Jouni 1 ; Paulin, Frédéric 2

1 Department of Mathematics and Statistics P.O. Box 35 40014 University of Jyväskylä FINLAND
2 Laboratoire de mathématique d’Orsay UMR 8628 CNRS Université Paris-Saclay 91405 ORSAY Cedex FRANCE
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     journal = {Annales math\'ematiques Blaise Pascal},
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Parkkonen, Jouni; Paulin, Frédéric. Rigidity, counting and equidistribution  of quaternionic Cartan chains. Annales mathématiques Blaise Pascal, Tome 28 (2021) no. 1, pp. 45-69. doi : 10.5802/ambp.399. http://www.numdam.org/articles/10.5802/ambp.399/

[1] Benoist, Yves; Oh, Hee Effective equidistribution of S-integral points on symmetric varieties, Ann. Inst. Fourier, Volume 62 (2012) no. 5, pp. 1889-1942 | DOI | Numdam | MR | Zbl

[2] Benoist, Yves; Quint, Jean-François Stationary measures and invariant subsets of homogeneous spaces II, J. Am. Math. Soc., Volume 26 (2013) no. 3, pp. 659-734 | DOI | MR | Zbl

[3] Besse, Arthur L. Einstein manifolds, Classics in Mathematics, Springer, 2008 | MR

[4] Biquard, Olivier Quaternionic contact structures, Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics (Roma, 1999), World Scientific, 2001, pp. 23-30 | DOI | Zbl

[5] Breulmann, Stefan; Helmke, Volker The covolume of quaternion groups on the four-dimensional hyperbolic space, Acta Arith., Volume 77 (1996) no. 1, pp. 9-21 | DOI | MR | Zbl

[6] Broise-Alamichel, Anne; Parkkonen, Jouni; Paulin, Frédéric Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation, Progress in Mathematics, 329, Birkhäuser, 2019

[7] Cartan, Élie Sur le groupe de la géométrie hypersphérique, Comment. Math. Helv., Volume 4 (1932), pp. 158-171 | DOI | Zbl

[8] Duke, William Rational points on the sphere, Ramanujan J., Volume 7 (2003) no. 1-3, pp. 235-239 | DOI | MR | Zbl

[9] Ellenberg, Jordan S.; Michel, Philippe; Venkatesh, Akshay Linnik’s ergodic method and the distribution of integer points on spheres, Automorphic representations and L-functions (Tata Institute of Fundamental Research Studies in Mathematics), Volume 22, Tata Institute of Fundamental Research, 2013, pp. 119-185 | MR | Zbl

[10] Goldman, William M. Complex hyperbolic geometry, Oxford Mathematical Monographs, Clarendon Press, 1999

[11] Gorodnik, Alexander; Maucourant, Francois Proximality and equidistribution on the Furstenberg boundary, Geom. Dedicata, Volume 113 (2005), pp. 197-213 | DOI | MR | Zbl

[12] Gromov, Mikhail Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry (Progress in Mathematics), Volume 144, Birkhäuser, 1996, pp. 79-323 | DOI | Zbl

[13] Gromov, Mikhail; Schoen, Richard Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math., Inst. Hautes Étud. Sci., Volume 76 (1992), pp. 165-246 | DOI | Numdam | Zbl

[14] Kamada, Hiroyuki; Nayatani, Shin Quaternionic CR geometry, Hokkaido Math. J., Volume 42 (2013) no. 2, pp. 159-207 | MR | Zbl

[15] Kim, Inkang Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds, J. Reine Angew. Math., Volume 704 (2015), pp. 85-133 | MR | Zbl

[16] Kim, Inkang; Parker, John R. Geometry of quaternionic hyperbolic manifolds, Math. Proc. Camb. Philos. Soc., Volume 135 (2003) no. 2, pp. 291-320 | MR | Zbl

[17] Mostow, George D. Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, 78, Princeton University Press, 1973

[18] Pansu, Pierre Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math., Volume 129 (1989) no. 1, pp. 1-60 | DOI | Zbl

[19] Parkkonen, Jouni; Paulin, Frédéric Spiraling spectra of geodesic lines in negatively curved manifolds, Math. Z., Volume 268 (2011) no. 1-2, pp. 101-142 Erratum in Math. Z., 276(3-4):1215–1216, 2014 | DOI | MR | Zbl

[20] Parkkonen, Jouni; Paulin, Frédéric Counting and equidistribution in Heisenberg groups, Math. Ann., Volume 367 (2017) no. 1-2, pp. 81-119 | DOI | MR | Zbl

[21] Parkkonen, Jouni; Paulin, Frédéric Counting common perpendicular arcs in negative curvature, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 3, pp. 900-938 | DOI | MR | Zbl

[22] Parkkonen, Jouni; Paulin, Frédéric Counting and equidistribution in quaternionic Heisenberg groups, Math. Proc. Camb. Philos. Soc. (2021) (To appear, https://doi.org/10.1017/S0305004121000426) | DOI

[23] Philippe, Zoé Invariants globaux des variétés hyperboliques quaternioniques, Ph. D. Thesis, Université de Bordeaux (France) (2016) (https://tel.archives-ouvertes.fr/tel-01661448)

[24] Shi, Yun Chain and -circle on quaternionic Heisenberg group and their properties [Chinese], Appl. Math., Ser. A (Chin. Ed.), Volume 31 (2016) no. 1, pp. 90-100 | MR | Zbl

[25] Vignéras, Marie-France Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, 800, Springer, 1980 | DOI | Zbl

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