The orbital counting problem for hyperconvex representations

Sambarino, Andrés

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1755-1797.

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 $PSL(2,ℝ)\times PSL(2,ℝ)$ induites par la choix de deux éléments de l’espace de Teichmüller de la surface et les representations dans la composante de Hitchin de $PSL(d,ℝ).$ Nous démontrons aussi, dans ce contexte, une propriété de melange pour le flot des chambres de Weyl.

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 $PSL(2,ℝ)\times PSL(2,ℝ),$ induced by choosing two points on the Teichmüller space of the surface and representations on the Hitchin component of $PSL(d,ℝ).$ We also prove a mixing property for the Weyl chamber flow in this setting.

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

[2] Benoist, Y. Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47 | DOI | MR | Zbl

[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.), Volume 26, Math. Soc. Japan, Tokyo, 2000, pp. 33-48 | MR | Zbl

[4] Bowen, Rufus Periodic orbits for hyperbolic flows, Amer. J. Math., Volume 94 (1972), pp. 1-30 | DOI | MR | Zbl

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

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

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

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

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

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

[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., Volume 24 (1988) no. 1, pp. 73-98 | EuDML | Numdam | MR | Zbl

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

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

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

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

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

[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.), Volume 13, Univ. Grenoble I, Saint-Martin-d’Hères, 1995, pp. 97-122 | EuDML | Numdam | Zbl

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

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

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

[21] Quint, J.-F. Groupes convexes cocompacts en rang supérieur, Geom. Dedicata, Volume 113 (2005), pp. 1-19 | DOI | MR | Zbl

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

[23] Quint, Jean-François Groupes de Schottky et comptage, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 2, pp. 373-429 | DOI | EuDML | Numdam | MR | Zbl

[24] Ratner, M. Markov partitions for Anosov flows on $n$-dimensional manifolds, Israel J. Math., Volume 15 (1973), pp. 92-114 | DOI | MR | Zbl

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

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

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

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

[29] Thirion, Xavier Sous-groupes discrets de $SL(d,ℝ)$ et equidistribution dans les espaces symétriques, Université de Tours (France) (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, Volume 137 (2009) no. 3, pp. 387-421 | EuDML | Numdam | MR | Zbl

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