Duke Mathematical Journal

Lyapunov spectrum of ball quotients with applications to commensurability questions

André Kappes and Martin Möller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors.

As a corollary, we complete the classification of commensurability classes of all presently known nonarithmetic ball quotients.

Article information

Duke Math. J., Volume 165, Number 1 (2016), 1-66.

Received: 6 November 2012
Revised: 2 July 2014
First available in Project Euclid: 14 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

Lyapunov spectrum nonarithmetric lattices period maps


Kappes, André; Möller, Martin. Lyapunov spectrum of ball quotients with applications to commensurability questions. Duke Math. J. 165 (2016), no. 1, 1--66. doi:10.1215/00127094-3165969. https://projecteuclid.org/euclid.dmj/1444828413

Export citation


  • [1] I. Bouw, The $p$-rank of ramified covers of curves, Compositio Math. 126 (2001), 295–322.
  • [2] I. Bouw, Pseudo-elliptic bundles, deformation data and the reduction of Galois covers, Habilitatsschrift, Fakultät Mathematik, Universität Duisburg-Essen, 2005.
  • [3] I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), 139–185.
  • [4] J. Carlson, S. Müller-Stach, and C. Peters, Period Mappings and Period Domains, Cambridge Stud. Adv. Math. 85, Cambridge Univ. Press, Cambridge, 2003.
  • [5] E. Cattani, A. Kaplan, and W. Schmid, Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), 457–535.
  • [6] P. B. Cohen and J. Wolfart, Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes, Ann. Sci. École Norm. Sup. (4) 26 (1993), 665–690.
  • [7] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, Berlin, 1970.
  • [8] P. Deligne, “Un théorème de finitude pour la monodromie” in Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984), Progr. Math. 67, Birkhäuser, Boston, 1987, 1–19.
  • [9] P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89.
  • [10] P. Deligne and G. D. Mostow, Commensurabilities among Lattices in $\mathrm{PU}(1,n)$, Ann. of Math. Stud. 132, Princeton Univ. Press, Princeton, 1993.
  • [11] M. Deraux, J. Parker, and J. Paupert, Census of the complex hyperbolic sporadic triangle groups, Exp. Math. 20 (2011), 467–486.
  • [12] M. Deraux, J. Parker, and J. Paupert, New non-arithmetic complex hyperbolic lattices, to appear in Invent. Math., preprint, arXiv:1401.0308v4 [math.GT].
  • [13] A. Eskin, M. Kontsevich, and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn. 5 (2011), 319–353.
  • [14] A. Eskin, M. Kontsevich, and A.Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 207–333.
  • [15] H. Esnault and E. Viehweg, Chern classes of Gauss-Manin bundles of weight 1 vanish, $K$-Theory 26 (2002), 287–305.
  • [16] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), 1–103.
  • [17] G. Forni, C. Matheus, and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Syst. 34 (2014), 353–408.
  • [18] G. Forni, C. Matheus, and A. Zorich, Zero Lyapunov exponents of the Hodge bundle, Comment. Math. Helv. 89 (2014), 489–535.
  • [19] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984.
  • [20] W. M. Goldman, Complex Hyperbolic Geometry, Oxford Math. Monogr., Oxford Univ. Press, New York, 1999.
  • [21] P. A. Griffiths, Periods of integrals on algebraic manifolds, III: Some global differential-geometric properties of the period mapping, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 125–180.
  • [22] J. Grivaux and P. Hubert, Les exposants de Liapounoff du flot de Teichmüller (d’après Eskin-Kontsevich-Zorich), Astérisque 361 (2014), 43–75, Séminaire Bourbaki, 2011/2012, nos. 1059–1065.
  • [23] B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. Math. (2) 173 (2003), 316–352.
  • [24] S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surv. Monogr. 39, Amer. Math. Soc., Providence, 1994.
  • [25] B. Hunt, Higher-dimensional ball quotients and the invariant quintic, Transform. Groups 5 (2000), 121–156.
  • [26] D. Huybrechts, Complex Geometry: An Introduction, Universitext, Springer, Berlin, 2005.
  • [27] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Wiley, New York, 1996.
  • [28] M. Kontsevich, “Lyapunov exponents and Hodge theory” in The Mathematical Beauty of Physics (Saclay, 1996), Adv. Ser. Math. Phys. 24, World Sci., River Edge, N.J., 1997, 318–332.
  • [29] M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory, preprint, arXiv:hep-th/9701164v1.
  • [30] S. Lang, Algebra, Grad. Texts in Math. 211, Springer, New York, 2002.
  • [31] A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. London Math. Soc. (3) 86 (2003), 358–396.
  • [32] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgb. (3) 17, Springer, Berlin, 1991.
  • [33] C. T. McMullen, Braid groups and Hodge theory, Math. Ann. 355 (2013), 893–946.
  • [34] C. T. McMullen, The Gauss–Bonnet theorem for cone manifolds and volumes of moduli spaces, preprint, 2013, http://www.math.harvard.edu/~ctm/papers/home/text/papers/gb/gb.pdf.
  • [35] D. B. McReynolds, Arithmetic lattices in $SU(n,1)$, preprint, 2011, http://www.math.uchicago.edu/~dmcreyn/Papers.html.
  • [36] M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5 (2011), 1–32.
  • [37] M. Möller, E. Viehweg, and K. Zuo, Stability of Hodge bundles and a numerical characterization of Shimura varieties, J. Differential Geom. 92 (2012), 71–151.
  • [38] G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), 171–276.
  • [39] G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106.
  • [40] G. D. Mostow, On discontinuous action of monodromy groups on the complex $n$-ball, J. Amer. Math. Soc. 1 (1988), 555–586.
  • [41] J. R. Parker, “Complex hyperbolic lattices” in Discrete Groups and Geometric Structures, Contemp. Math. 501, Amer. Math. Soc., Providence, 2009, 1–42.
  • [42] J. Paupert, Unfaithful complex hyperbolic triangle groups, III: Arithmeticity and commensurability, Pacific J. Math. 245 (2010), 359–372.
  • [43] C. A. M. Peters and J. H. M. Steenbrink, “Monodromy of variations of Hodge structure” in Monodromy and Differential Equations (Moscow, 2001), Appl. Math. 75 (2003), 183–194.
  • [44] H. L. Royden, The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980), 547–558.
  • [45] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • [46] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58.
  • [47] J. K. Sauter, Jr, Isomorphisms among monodromy groups and applications to lattices in $\mathrm{PU}(1,2)$, Pacific J. Math. 146 (1990), 331–384.
  • [48] W. Schmid, Variation of Hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973), 211–319.
  • [49] H. Shiga, On holomorphic mappings of complex manifolds with ball model, J. Math. Soc. Japan 56 (2004), 1087–1107.
  • [50] W. P. Thurston, “Shapes of polyhedra and triangulations of the sphere” in The Epstein Birthday Schrift, Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry, 1998, 511–549.
  • [51] E. Viehweg and K. Zuo, Arakelov inequalities and the uniformization of certain rigid Shimura varieties, J. Differential Geom. 77 (2007), 291–352.
  • [52] A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn. 6 (2012), 405–426.
  • [53] M. Yoshida, Fuchsian Differential Equations, Aspects Math. E11, Vieweg, Braunschweig, 1987.
  • [54] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics and Geometry, Volume 1: On Random Matrices, Zeta Functions and Dynamical Systems, Springer, Berlin, 2006, 439–586.