The orbital counting problem for hyperconvex representations
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1755-1797.

We give a precise counting result on the symmetric space of a connected noncompact real-algebraic semisimple Lie group G, for a class of discrete subgroups of G that contains, for example, representations of a surface group on PSL(2,)×PSL(2,), induced by choosing two points on the Teichmüller space of the surface and representations on the Hitchin component of PSL(d,). We also prove a mixing property for the Weyl chamber flow in this setting.

Nous trouvons un asymptotique pour le comptage orbitale dans l’espace symétrique d’un groupe de Lie connexe, réel-algébrique, semisimple et non-compact G, pour une classe des sous groupes discrets de G qui contient, par exemple, representations d’un groupe de surface dans PSL(2,)×PSL(2,) induites par la choix de deux éléments de l’espace de Teichmüller de la surface et les representations dans la composante de Hitchin de PSL(d,). Nous démontrons aussi, dans ce contexte, une propriété de melange pour le flot des chambres de Weyl.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.2973
Classification: 22E40,  37D20
Keywords: Lie groups, higher rank geometries, Hitchin representations
@article{AIF_2015__65_4_1755_0,
     author = {Sambarino, Andr\'es},
     title = {The orbital counting problem for hyperconvex representations},
     journal = {Annales de l'Institut Fourier},
     pages = {1755--1797},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {4},
     year = {2015},
     doi = {10.5802/aif.2973},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2973/}
}
TY  - JOUR
TI  - The orbital counting problem for hyperconvex representations
JO  - Annales de l'Institut Fourier
PY  - 2015
DA  - 2015///
SP  - 1755
EP  - 1797
VL  - 65
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2973/
UR  - https://doi.org/10.5802/aif.2973
DO  - 10.5802/aif.2973
LA  - en
ID  - AIF_2015__65_4_1755_0
ER  - 
%0 Journal Article
%T The orbital counting problem for hyperconvex representations
%J Annales de l'Institut Fourier
%D 2015
%P 1755-1797
%V 65
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2973
%R 10.5802/aif.2973
%G en
%F AIF_2015__65_4_1755_0
Sambarino, Andrés. The orbital counting problem for hyperconvex representations. Annales de l'Institut Fourier, Volume 65 (2015) no. 4, pp. 1755-1797. doi : 10.5802/aif.2973. https://aif.centre-mersenne.org/articles/10.5802/aif.2973/

[1] Babillot, M. Théorie du renouvellement pour des chaînes semi-markoviennes transientes, Ann. Inst. H. Poincaré Probab. Statist., Tome 24 (1988) no. 4, pp. 507-569 | Numdam | MR: 978023 | Zbl: 0681.60095

[2] Benoist, Y. Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Tome 7 (1997) no. 1, pp. 1-47 | Article | MR: 1437472 | Zbl: 0947.22003

[3] Benoist, Yves Propriétés asymptotiques des groupes linéaires. II, Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997) (Adv. Stud. Pure Math.) Tome 26, Math. Soc. Japan, Tokyo, 2000, pp. 33-48 | MR: 1770716 | Zbl: 0960.22012

[4] Bowen, Rufus Periodic orbits for hyperbolic flows, Amer. J. Math., Tome 94 (1972), pp. 1-30 | Article | MR: 298700 | Zbl: 0254.58005

[5] Bowen, Rufus Symbolic dynamics for hyperbolic flows, Amer. J. Math., Tome 95 (1973), pp. 429-460 | Article | Zbl: 0282.58009

[6] Bowen, Rufus; Ruelle, David The ergodic theory of Axiom A flows, Invent. Math., Tome 29 (1975) no. 3, pp. 181-202 | Article | MR: 380889 | Zbl: 0311.58010

[7] Duke, W.; Rudnick, Z.; Sarnak, P. Density of integer points on affine homogeneous varieties, Duke Math. J., Tome 71 (1993) no. 1, pp. 143-179 | Article | MR: 1230289 | Zbl: 0798.11024

[8] Efremovič, V. A.; Tihomirova, E. S. Equimorphisms of hyperbolic spaces, Izv. Akad. Nauk SSSR Ser. Mat., Tome 28 (1964), pp. 1139-1144 | MR: 169121

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

[10] Sur les groupes hyperboliques d’après Mikhael Gromov (Ghys, É.; de la Harpe, P., eds.), Progress in Mathematics, Tome 83, Birkhäuser Boston, Inc., Boston, MA, 1990, xii+285 pages (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | Article | MR: 1086648 | Zbl: 0731.20025

[11] Guivarc’h, Y.; Hardy, J. Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré Probab. Statist., Tome 24 (1988) no. 1, pp. 73-98 | EuDML: 77319 | Numdam | MR: 937957 | Zbl: 0649.60041

[12] Guivarc’h, Yves; Ji, Lizhen; Taylor, J. C. Compactifications of symmetric spaces, Progress in Mathematics, Tome 156, Birkhäuser Boston, Inc., Boston, MA, 1998, xiv+284 pages | MR: 1633171 | Zbl: 1053.31006

[13] Howe, Roger E.; Moore, Calvin C. Asymptotic properties of unitary representations, J. Funct. Anal., Tome 32 (1979) no. 1, pp. 72-96 | Article | MR: 533220 | Zbl: 0404.22015

[14] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, xiv+247 pages (Graduate Texts in Mathematics, No. 21) | MR: 396773 | Zbl: 0471.20029

[15] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, Tome 54, Cambridge University Press, Cambridge, 1995, xviii+802 pages (With a supplementary chapter by Katok and Leonardo Mendoza) | Article | MR: 1326374 | Zbl: 0878.58020

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

[17] Ledrappier, François Structure au bord des variétés à courbure négative, Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994–1995 (Sémin. Théor. Spectr. Géom.) Tome 13, Univ. Grenoble I, Saint-Martin-d’Hères, 1995, pp. 97-122 | EuDML: 114385 | Numdam | Zbl: 0931.53005

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

[19] Margulis, G. A. Certain applications of ergodic theory to the investigation of manifolds of negative curvature, Funkcional. Anal. i Priložen., Tome 3 (1969) no. 4, p. 89-90 | MR: 257933 | Zbl: 0207.20305

[20] Quint, J.-F. Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal., Tome 12 (2002) no. 4, pp. 776-809 | Article | MR: 1935549 | Zbl: 1169.22300

[21] Quint, J.-F. Groupes convexes cocompacts en rang supérieur, Geom. Dedicata, Tome 113 (2005), pp. 1-19 | Article | MR: 2171296 | Zbl: 1077.22016

[22] Quint, Jean-François Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv., Tome 77 (2002) no. 3, pp. 563-608 | Article | MR: 1933790 | Zbl: 1010.22018

[23] Quint, Jean-François Groupes de Schottky et comptage, Ann. Inst. Fourier (Grenoble), Tome 55 (2005) no. 2, pp. 373-429 http://aif.cedram.org/item?id=AIF_2005__55_2_373_0 | Article | EuDML: 116195 | Numdam | MR: 2147895 | Zbl: 1087.22010

[24] Ratner, M. Markov partitions for Anosov flows on n-dimensional manifolds, Israel J. Math., Tome 15 (1973), pp. 92-114 | Article | MR: 339282 | Zbl: 0269.58010

[25] Roblin, Thomas Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) (2003) no. 95, vi+96 pages | Numdam | Zbl: 1056.37034

[26] Sambarino, A. Hyperconvex representations and exponential growth, Ergodic Theory Dynam. Systems, Tome 34 (2014) no. 3, pp. 986-1010 | Article | MR: 3199802 | Zbl: 1308.37014

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

[28] Shub, Michael Global stability of dynamical systems, Springer-Verlag, New York, 1987, xii+150 pages (With the collaboration of Albert Fathi and Rémi Langevin, Translated from the French by Joseph Christy) | Article | MR: 869255 | Zbl: 0606.58003

[29] Thirion, Xavier Sous-groupes discrets de SL(d,) et equidistribution dans les espaces symétriques (2007) (Ph. D. Thesis)

[30] Thirion, Xavier Propriétés de mélange du flot des chambres de Weyl des groupes de ping-pong, Bull. Soc. Math. France, Tome 137 (2009) no. 3, pp. 387-421 | EuDML: 272477 | Numdam | MR: 2574089 | Zbl: 1183.22005

[31] Tits, J. Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., Tome 247 (1971), pp. 196-220 | EuDML: 151104 | MR: 277536 | Zbl: 0227.20015

Cited by Sources: