Counting problems for special-orthogonal Anosov representations

Carvajales, León

Annales de l'Institut Fourier, Tome 70 (2020) no. 3, pp. 1199-1257.

Pour des entiers positifs $p$ et $q$ soit $G:=\mathrm{PSO}(p,q)$ 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 ${ℍ}^{p,q-1}$. Soit $S\subset X_G$ une copie totalement géodésique de $X_{\mathrm{PSO}(p,q-1)}$. 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 ${ℍ}^{p,q-1}$, 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 $G:=\mathrm{PSO}(p,q)$ 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 ${ℍ}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic copy of $X_{\mathrm{PSO}(p,q-1)}$. 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 ${ℍ}^{p,q-1}$, as the asymptotics of the amount of space-like geodesic segments of maximum length $t$ in the orbit of a point.

Classification : 22E40, 53C35
Keywords: Anosov representations, counting, symmetric spaces
Mot clés : Représentations d’Anosov, comptage, espaces symétriques

[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

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

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

[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

[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 | Zbl

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

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

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

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

[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

[11] Busemann, Herbert Spaces with non-positive curvature, Acta Math., Volume 80 (1948), pp. 259-311

[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

[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

[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

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

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

[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

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

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

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

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

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

[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

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

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

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

[29] Knapp, Anthony Lie groups beyond an introduction, Progress in Mathematics, 140, Springer, 1996

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

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

[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

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

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

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

[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

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

[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

[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

[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

[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

[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

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

[44] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 1998

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

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

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

[48] Quint, Jean-François Groupes de Schottky et comptage, Ann. Inst. Fourier, Volume 55 (2005) no. 2, pp. 373-429

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

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

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

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

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