Abstract
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-16-CE40-0025
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 338644254
Funding source: European Research Council
Award Identifier / Grant number: 614733
Funding statement: Andrés Sambarino was partially financed by ANR DynGeo ANR-16-CE40-0025. Beatrice Pozzetti and Anna Wienhard acknowledge funding by the Deutsche Forschungsgemeinschaft Project number 338644254 within the Priority Program SPP 2026 “Geometry at Infinity”. Anna Wienhard acknowledges funding by the European Research Council under ERC-Consolidator grant 614733, and by the Klaus-Tschira-Foundation.
References
[1] B. Bárány, M. Hochman and A. Rapaport, Hausdorff dimension of planar self-affine sets and measures, Invent. Math. 216 (2019), no. 3, 601–659. 10.1007/s00222-018-00849-ySearch in Google Scholar
[2] Y. Benoist, Convexes divisibles. I, Algebraic groups and arithmetic, Tata Institute of Fundamental Research, Mumbai (2004), 339–374. 10.1016/S0764-4442(01)01878-XSearch in Google Scholar
[3] J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z. 263 (2009), no. 1, 221–231. 10.1007/s00209-009-0494-ySearch in Google Scholar
[4] J. Bochi, R. Potrie and A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. (JEMS) 21 (2019), no. 11, 3343–3414. 10.4171/JEMS/905Search in Google Scholar
[5]
M. Bourdon,
Structure conforme au bord et flot géodésique d’un
[6] M. Bridgeman, R. Canary, F. Labourie and A. Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015), no. 4, 1089–1179. 10.1007/s00039-015-0333-8Search in Google Scholar
[7] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren Math. Wiss. 319, Springer, Berlin 1999. 10.1007/978-3-662-12494-9Search in Google Scholar
[8] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972), 5–251. 10.1007/BF02715544Search in Google Scholar
[9] M. Burger and A. Iozzi, A measurable Cartan theorem and applications to deformation rigidity in complex hyperbolic geometry, Pure Appl. Math. Q. 4 (2008), 181–202. 10.4310/PAMQ.2008.v4.n1.a8Search in Google Scholar
[10] J. Chen and Y. Pesin, Dimension of non-conformal repellers: A survey, Nonlinearity 23 (2010), no. 4, 93–114. 10.1088/0951-7715/23/4/R01Search in Google Scholar
[11] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math. 1441, Springer, Berlin 1990. 10.1007/BFb0084913Search in Google Scholar
[12] S. Crovisier and R. Potrie, Introduction to partially hyperbolic dynamics, Lectures Notes, Trieste, 2015. Search in Google Scholar
[13] J. Danciger, F. Guéritaud and F. Kassel, Proper affine actions for right-angled Coxeter groups, Duke Math. J. 169 (2020), 2231–2280. 10.1215/00127094-2019-0084Search in Google Scholar
[14] J. Danciger and T. Zhang, Affine actions with Hitchin linear part, Geom. Funct. Anal. 29 (2019), no. 5, 1369–1439. 10.1007/s00039-019-00511-6Search in Google Scholar
[15] L. Dufloux, Dimension de Hausdorff des ensembles limites, preprint (2015), https://hal.archives-ouvertes.fr/tel-01293924/document. Search in Google Scholar
[16] L. Dufloux, Hausdorff dimension of limit sets, Geom. Dedicata 191 (2017), 1–35. 10.1007/s10711-017-0240-2Search in Google Scholar
[17] V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. 10.1007/s10240-006-0039-4Search in Google Scholar
[18] D. Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. 10.2307/2946597Search in Google Scholar
[19] O. Glorieux, D. Monclair and N. Tholozan, Hausdorff dimension of limit set for projective Anosov groups, preprint (2019), https://arxiv.org/abs/1902.01844. Search in Google Scholar
[20] F. Guéritaud, O. Guichard, F. Kassel and A. Wienhard, Anosov representations and proper actions, Geom. Topol. 21 (2017), 485–584. 10.2140/gt.2017.21.485Search in Google Scholar
[21] O. Guichard, Personal communication. Search in Google Scholar
[22] O. Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, J. Differential Geom. 80 (2008), no. 3, 391–431. 10.4310/jdg/1226090482Search in Google Scholar
[23] O. Guichard and A. Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012), no. 2, 357–438. 10.1007/s00222-012-0382-7Search in Google Scholar
[24] P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Séminaire Bourbaki. Volume 2007/2008. Exposés 982-996, Astérisque 26, Société Mathématique de France, Paris (2009), 321–362, Exp. No. 993. Search in Google Scholar
[25] J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math. 9, Springer, New York 1972. 10.1007/978-1-4612-6398-2Search in Google Scholar
[26] D. Johnson and J. J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis (New Haven 1984), Progr. Math. 67, Birkhäuser, Boston (1987), 48–106. 10.1007/978-1-4899-6664-3_3Search in Google Scholar
[27] M. Kapovich, B. Leeb and J. Porti, A Morse lemma for quasigeodesics in symmetric spaces and euclidean buildings, preprint (2014), https://arxiv.org/abs/1411.4176v1. 10.2140/gt.2018.22.3827Search in Google Scholar
[28] M. Kapovich, B. Leeb and J. Porti, Morse actions of discrete groups on symmetric space, preprint (2014), http://front.math.ucdavis.edu/1403.7671. Search in Google Scholar
[29] B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032. 10.1007/b94535_11Search in Google Scholar
[30] F. Labourie, Fuchsian affine actions of surface groups, J. Differential Geom. 59 (2001), no. 1, 15–31. 10.4310/jdg/1090349279Search in Google Scholar
[31] F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), no. 1, 51–114. 10.1007/s00222-005-0487-3Search in Google Scholar
[32] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. 10.2307/1971328Search in Google Scholar
[33] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University, Cambridge 1995. 10.1017/CBO9780511626302Search in Google Scholar
[34] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3) 17, Springer, Berlin 1991. 10.1007/978-3-642-51445-6Search in Google Scholar
[35] R. Potrie and A. Sambarino, Eigenvalues and entropy of a Hitchin representation, Invent. Math. 209 (2017), no. 3, 885–925. 10.1007/s00222-017-0721-9Search in Google Scholar
[36]
M. B. Pozzetti,
Maximal representations of complex hyperbolic lattices into
[37] J.-F. Quint, Cônes limites des sous-groupes discrets des groupes réductifs sur un corps local, Transform. Groups 7 (2002), no. 3, 247–266. 10.1007/s00031-002-0013-2Search in Google Scholar
[38] J.-F. Quint, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002), no. 3, 563–608. 10.1007/s00014-002-8352-0Search in Google Scholar
[39] J.-F. Quint, Mesures de Patterson–Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002), no. 4, 776–809. 10.1007/s00039-002-8266-4Search in Google Scholar
[40] M. Sambarino, A (short) survey on dominated splittings, Mathematical Congress of the Americas, Contemp. Math. 656, American Mathematical Society, Providence (2016), 149–183. 10.1090/conm/656/13105Search in Google Scholar
[41] F. Stecker and N. Treib, Domains of discontinuity in oriented flag manifolds, preprint (2018), https://arxiv.org/abs/1806.04459. 10.1112/jlms.12602Search in Google Scholar
[42] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202. 10.1007/BF02684773Search in Google Scholar
[43] P. Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157–187. Search in Google Scholar
[44] C. Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4965–5005. 10.1090/S0002-9947-96-01614-5Search in Google Scholar
[45] T. Zhang and A. Zimmer, Regularity of limit sets of Anosov representations, preprint (2018), https://arxiv.org/abs/1903.11021. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
