Counting problems for special-orthogonal Anosov representations  [ Problèmes de comptage pour les représentations d’Anosov dans le groupe spécial orthogonal ]
Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1199-1257.

Pour des entiers positifs p et q soit G:=PSO(p,q) le groupe projectif spécial-orthogonal indéfini de signature (p,q). Nous étudions des problèmes de comptage dans l’espace symétrique Riemannien X G de G et dans l’espace hyperbolique pseudo-Riemannien p,q-1 . Soit SX G une copie totalement géodésique de X PSO(p,q-1) . Nous examinons l’orbite de S sous l’action d’un sous-groupe de G de type projectivement Anosov. Pour certains choix d’une telle copie géodésique, nous montrons que le nombre de points dans cette orbite qui se trouvent à une distance maximale t de S est fini et asymptotiquement purement exponentiel lorsque t tend vers l’infini. Nous fournissons une interprétation de ce résultat dans p,q-1 , comme l’asymptotique de la quantité de segments géodésiques de type espace de longueur maximale t dans l’orbite d’un point.

For positive integers p and q let G:=PSO(p,q) be the projective indefinite special-orthogonal group of signature (p,q). We study counting problems in the Riemannian symmetric space X G of G and in the pseudo-Riemannian hyperbolic space p,q-1 . Let SX G be a totally geodesic copy of X PSO(p,q-1) . We look at the orbit of S under the action of a projective Anosov subgroup of G. For certain choices of such a geodesic copy we show that the number of points in this orbit which are at distance at most t from S is finite and asymptotic to a purely exponential function as t goes to infinity. We provide an interpretation of this result in p,q-1 , as the asymptotics of the amount of space-like geodesic segments of maximum length t in the orbit of a point.

Reçu le : 2018-12-05
Révisé le : 2019-09-02
Accepté le : 2019-09-18
Publié le : 2020-06-26
DOI : https://doi.org/10.5802/aif.3333
Classification : 22E40,  53C35
Mots clés: Représentations d’Anosov, comptage, espaces symétriques
@article{AIF_2020__70_3_1199_0,
     author = {Carvajales, Le\'on},
     title = {Counting problems for special-orthogonal Anosov representations},
     journal = {Annales de l'Institut Fourier},
     pages = {1199--1257},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.5802/aif.3333},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1199_0/}
}
Carvajales, León. Counting problems for special-orthogonal Anosov representations. Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1199-1257. doi : 10.5802/aif.3333. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1199_0/

[1] Babillot, Martine Points entiers et groupes discrets : de l’analyse aux systèmes dynamiques, Rigidité, groupe fondamental et dynamique (Panoramas et Synthèses) Volume 13, Société Mathématique de France, 2002, pp. 1-119 | MR 1993148 | Zbl 1077.11071

[2] Benoist, Yves Actions propres sur les espaces homogènes réductifs, Ann. Math., Volume 144 (1996) no. 2, pp. 315-347 | Article | Zbl 0868.22013

[3] Benoist, Yves Propriétés asymptotiques des groupes linéares, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47 | Article

[4] Benoist, Yves Propriétés asymptotiques des groupes linéares II, Analysis on homogeneous spaces and representation theory of Lie groups (Okayama–Kyoto, 1997) (Advanced Studies in Pure Mathematics) Volume 26, Mathematical Society of Japan, 2000, pp. 33-48 | Article | Zbl 0960.22012

[5] Benoist, Yves Convexes divisibles I, Algebraic groups and arithmetic. Proceedings of the international conference, (Mumbai, India, December 17–22, 2001) (Studies in Mathematics. Tata Institute of Fundamental Research) Volume 17, Narosa Publishing House/Published for the Tata Institute of Fundamental Research, 2004, pp. 339-374 | MR 2094116 | Zbl 1084.37026

[6] Bochi, Jairo; Potrie, Rafael; Sambarino, Andrés Anosov representations and dominated splittings, J. Eur. Math. Soc., Volume 21 (2019) no. 11, pp. 3343-3414 | Article | MR 4012341 | Zbl 1429.22011

[7] Bowditch, Brian Convergence groups and configuration spaces, Geometric group theory down under. Proceedings of a Special Year in Geometric Group Theory, Canberra, Australia (1999), pp. 23-54 | Zbl 0952.20032

[8] Bowen, Rufus Periodic orbits of hyperbolic flows, Am. J. Math., Volume 94 (1972), pp. 1-30 | Article | MR 298700 | Zbl 0254.58005

[9] Bowen, Rufus; Marcus, Brian Unique ergodicity for horocycle foliations, Isr. J. Math., Volume 26 (1977) no. 1, pp. 43-67 | Article | MR 451307 | Zbl 0346.58009

[10] Bridgeman, Martin; Canary, Richard; Labourie, François; Sambarino, Andrés The pressure metric for Anosov representations, Geom. Funct. Anal., Volume 25 (2015) no. 4, pp. 1089-1179 | Article | MR 3385630 | Zbl 1360.37078

[11] Busemann, Herbert Spaces with non-positive curvature, Acta Math., Volume 80 (1948), pp. 259-311 | Article | MR 29531 | Zbl 0038.10005

[12] Constantine, Dave; Lafont, Jean-François; Thompson, Dan Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows, J. Éc. Polytech., Math, Volume 7 (2020), pp. 201-231 | Article | MR 4054334

[13] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Convex cocompact actions in real projective geometry (2017) (https://arxiv.org/abs/1704.08711)

[14] Danciger, Jeffrey; Guéritaud, François; Kassel, Fanny Convex cocompactness in pseudo-Riemannian hyperbolic spaces, Geom. Dedicata, Volume 192 (2018) no. 1, pp. 87-126 | Article | MR 3749424 | Zbl 1428.53078

[15] Duke, William; Rudnick, Zeev; Sarnak, Peter Density of integer points on affine homogeneous varieties, Duke Math. J., Volume 71 (1993) no. 1, pp. 143-179 | Article | MR 1230289 | Zbl 0798.11024

[16] Eskin, Alex; McMullen, Curtis Mixing, counting and equidistribution in Lie groups, Duke Math. J., Volume 71 (1993) no. 1, pp. 181-209 | Article | MR 1230290 | Zbl 0798.11025

[17] Ghys, Étienne; Harpe, Pierre de la Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, Volume 83, Springer, 1990 | Zbl 0731.20025

[18] Glorieux, Olivier; Monclair, Daniel Critical exponent and Hausdorff dimension for quasi-Fuchsian AdS manifolds (2016) (https://arxiv.org/abs/1606.05512)

[19] Gromov, Mikhael Hyperbolic groups, Essays in group theory (Mathematical Sciences Research Institute Publications) Volume 8, Springer, 1987 | Article | MR 919829

[20] Guichard, Olivier; Guéritaud, François; Kassel, Fanny; Wienhard, Anna Anosov representations and proper actions, Geom. Topol., Volume 21 (2017) no. 1, pp. 485-584 | MR 3608719 | Zbl 1373.37095

[21] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | Article | MR 2981818 | Zbl 1270.20049

[22] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Academic Press Inc., 1978 | Zbl 0451.53038

[23] Horn, Roger; Johnson, Charles Matrix analysis, Cambridge University Press, 1985 | Zbl 0576.15001

[24] Kapovich, Michael; Leeb, Bernhard; Porti, Joan Anosov subgroups: Dynamical and geometric characterizations, Eur. J. Math., Volume 3 (2017) no. 4, pp. 808-898 | Article | MR 3736790 | Zbl 1403.22013

[25] Kapovich, Michael; Leeb, Bernhard; Porti, Joan A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings, Geom. Topol., Volume 22 (2018) no. 7, pp. 3827-3923 | Article | MR 3890767 | Zbl 06997379

[26] Kassel, Fanny Geometric structures and representations of discrete groups, Proceedings of the International Congress of Mathematicians (ICM 2018), Volume 1 (2018), pp. 1115-1151 | MR 3966802

[27] Kassel, Fanny; Kobayashi, Toshiyuki Poincaré series for non-Riemannian locally symmetric spaces, Adv. Math., Volume 287 (2016), pp. 123-236 | Article | Zbl 1333.53073

[28] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and Its Applications, Volume 54, Cambridge University Press, 1995, xviii+802 pages | Article | MR 1326374 | Zbl 0878.58020

[29] Knapp, Anthony Lie groups beyond an introduction, Progress in Mathematics, Volume 140, Springer, 1996 | MR 1399083 | Zbl 0862.22006

[30] Labourie, François Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114 | Article | MR 2221137 | Zbl 1103.32007

[31] Ledrappier, François Structure au bord des variétés à courbure négative, Séminaire de Théorie spectrale et géométrie de Grenoble, Volume 13 (19954), pp. 97-122 | Article | Zbl 0931.53005

[32] Lee, Min; Oh, Hee Effective circle count for Apollonian packings and closed horospheres, Geom. Funct. Anal., Volume 23 (2013) no. 2, pp. 580-621 | MR 3053757 | Zbl 1276.52019

[33] Livšic, A N Cohomology of dynamical systems, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 36 (1972), pp. 1296-1320 | MR 334287

[34] Margulis, Grigori Applications of ergodic theory to the investigation of manifolds with negative curvature, Funct. Anal. Appl., Volume 3 (1969), p. 335-336 | Article | MR 257933

[35] Margulis, Grigori Certain measures associated with U-flows on compact manifolds, Funct. Anal. Appl., Volume 4 (1970), pp. 55-67 | Article | Zbl 0245.58003

[36] Matsuki, Toshihiko The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan, Volume 31 (1979) no. 2, pp. 331-357 | Article | MR 527548

[37] Mineyev, Igor Flows and joins of metric spaces, Geom. Topol., Volume 9 (2005), pp. 403-482 | Article | MR 2140987 | Zbl 1137.37314

[38] Mohammadi, Amir; Oh, Hee Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc., Volume 17 (2015) no. 4, pp. 837-897 | Article | MR 3336838 | Zbl 1391.11137

[39] Oh, Hee; Shah, Nimish The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math., Volume 187 (2012) no. 1, pp. 1-35 | Article | MR 2874933 | Zbl 1235.52033

[40] Oh, Hee; Shah, Nimish Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Am. Math. Soc., Volume 26 (2013) no. 2, pp. 511-562 | Article | MR 3011420 | Zbl 1334.22011

[41] Oh, Hee; Shah, Nimish Counting visible circles on the sphere and Kleinian groups, Proceedings of the conference on “Geometry, Topology and Dynamics in negative curvature” (London Mathematical Society Lecture Note Series) Volume 425, Cambridge University Press, 2016, pp. 272-288 | Article | MR 3497263 | Zbl 1368.52013

[42] Parry, William; Pollicott, Mark An analogue of the prime number theorem and closed orbits of Axiom A flows, Ann. Math., Volume 118 (1983), pp. 573-591 | Article | MR 727704 | Zbl 0537.58038

[43] Patterson, Samuel James The limit set of a Fuchsian group, Acta Math., Volume 136 (1976), pp. 241-273 | Article | MR 450547

[44] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, Volume 171, Springer, 1998 | MR 1480173 | Zbl 0914.53001

[45] Phillips, Ralph; Rudnick, Zeev The circle problem in the hyperbolic plane, J. Funct. Anal., Volume 121 (1994) no. 1, pp. 78-116 | Article | MR 1270589 | Zbl 0812.11035

[46] Pollicott, Mark Symbolic dynamics for Smale flows, Am. J. Math., Volume 109 (1987) no. 1, pp. 183-200 | Article | MR 878205 | Zbl 0628.58042

[47] Quint, Jean-François Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal., Volume 12 (2002) no. 4, pp. 776-809 | Article | Zbl 1169.22300

[48] Quint, Jean-François Groupes de Schottky et comptage, Ann. Inst. Fourier, Volume 55 (2005) no. 2, pp. 373-429 | Article | Numdam | MR 2147895 | Zbl 1087.22010

[49] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr., Nouv. Sér., Volume 95 (2003), p. vi+96 | Numdam | MR 2057305 | Zbl 1056.37034

[50] Sambarino, Andrés Quantitative properties of convex representations, Comment. Math. Helv., Volume 89 (2014) no. 2, pp. 443-488 | Article | MR 3229035 | Zbl 1295.22016

[51] Sambarino, Andrés The orbital counting problem for hyperconvex representations, Ann. Inst. Fourier, Volume 65 (2015) no. 4, pp. 1755-1797 | Article | Numdam | MR 3449196 | Zbl 1354.22016

[52] Schlichtkrull, Henrik Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, Birkhäuser, 1984 no. 49 | Zbl 0555.43002

[53] Wienhard, Anna An invitation to higher Teichmüller theory, Proceedings of the International Congress of Mathematicians (ICM 2018) (2018), pp. 1013-1039 | Article