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Communications in Mathematical Physics

, Volume 340, Issue 2, pp 433–469 | Cite as

Lyapunov Exponents for Surface Group Representations

  • Bertrand Deroin
  • Romain DujardinEmail author
Article

Abstract

Let \({(\rho_\lambda)_{\lambda \in \Lambda}}\) be a holomorphic family of representations of a surface group \({\pi_1(S)}\) into \({\mathrm{PSL}(2, \mathbb{C})}\), where S is a topological (possibly punctured) surface with negative Euler characteristic. Given a structure of Riemann surface of finite type on S we construct a bifurcation current on the parameter space Λ, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the dd c of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincaré metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in Λ. For instance, the random hypersurfaces of Λ defined by the condition that a random closed geodesic on S is mapped under ρ λ to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work (Deroin and Dujardin, Invent Math 190:57–118, 2012), and on a careful control of a discretization procedure of the Brownian motion.

Keywords

Brownian Motion Riemann Surface Lyapunov Exponent Heat Kernel Spherical Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Al.
    Alvarez, S.: Discretization of harmonic measures for foliated bundles. C. R. Math. Acad. Sci. Paris 350, 621–626 (2102)Google Scholar
  2. An.
    Ancona, A.: Théorie du potentiel sur les graphes et les variétés. École d’été de Probabilités de Saint-Flour XVIII, pp. 1–112, Lecture Notes in Math., vol. 1427. Springer, Berlin (1990)Google Scholar
  3. Ao.
    Aoun R.: Random subgroups of linear groups are free. Duke Math. J. 160, 117–173 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  4. BaLe.
    Ballmann, W., Ledrappier, F.: Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), pp. 77–92, Sémin. Congr., vol. 1. Soc. Math. France, Paris (1996)Google Scholar
  5. Bea.
    Beardon A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, vol. 91. Springer, New York (1983)Google Scholar
  6. Bo.
    Bonahon F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92(1), 139–162 (1988)zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. BGM.
    Bonatti C., Gómez-Mont X., Viana M.: Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, 579–624 (2003)zbMATHCrossRefADSGoogle Scholar
  8. BL.
    Bougerol P., Lacroix J.: Products of Random Matrices with Applications to Schrödinger operators. Progress in Probability and Statistics, vol. 8. Birkhäuser Boston, Inc, Boston (1985)CrossRefGoogle Scholar
  9. CC.
    Candel A., Conlon L.: Foliations. II. Graduate Studies in Mathematics, vol. 60. American Mathematical Society, Providence (2003)Google Scholar
  10. CK.
    Carmona R., Klein A.: Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Probab. 11(3), 648–655 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  11. CS.
    Culler M., Shalen P.B.: Varieties of group representations and splittings of 3-manifolds. Ann. Math. 117(2), 109–146 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  12. Da.
    Davies E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  13. DD1.
    Deroin B., Dujardin R.: Random walks, Kleinian groups, and bifurcation currents. Invent. Math. 190, 57–118 (2012)zbMATHMathSciNetCrossRefADSGoogle Scholar
  14. DD2.
    Deroin, B., Dujardin, R.: Complex projective structures Lyapunov exponent, degree and harmonic measure (2013, preprint). arXiv:1308.0541
  15. DDp.
    Deroin, B., Christophe D.: Topology and dynamics of laminations in surfaces of general type. J. Am. Math. Soc. (2015, to appear)Google Scholar
  16. DKN.
    Deroin B., Kleptsyn V., Andrés N.: On the question of ergodicity for minimal group actions on the circle. Mosc. Math. J. 9, 263–303 (2009)zbMATHMathSciNetGoogle Scholar
  17. Duj.
    Dujardin, R.: Bifurcation currents and equidistribution on parameter space. In: Frontiers in Complex Dynamics, pp. 515–566, Princeton Math. Ser., vol. 51. Princeton University Press, Princeton (2014)Google Scholar
  18. Dum.
    Dumas, D.: Complex projective structures. Handbook of Teichmüller theory, vol. II, pp. 455–508. Eur. Math. Soc., Zürich (2009)Google Scholar
  19. Dy.
    Dynkin, E.B.: Markov Processes I. Academic Press, New York; Springer, Berlin (1965)Google Scholar
  20. EM.
    Eskin A., McMullen C.T.: Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71, 181–209 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  21. FL.
    Franchi J., Le Jan Y.: Hyperbolic Dynamics and Brownian Motion: An Introduction. Oxford Mathematical Monographs. Oxford University Press, Oxford (2012)CrossRefGoogle Scholar
  22. F1.
    Furstenberg H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  23. F2.
    Furstenberg, H.: Random walks and discrete subgroups of Lie groups. 1971 Advances in Probability and Related Topics, vol. 1, pp. 1–63. Dekker, New YorkGoogle Scholar
  24. Gri.
    Grigor’yan A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. (N.S.) 36, 135–249 (1999)zbMATHCrossRefGoogle Scholar
  25. Gru.
    Gruet J.-C.: On the length of the homotopic brownian word in the thrice punctured sphere. Probab. Theory Relat. Fields 111(4), 489–516 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  26. Gu.
    Guivarc’h Y.: On contraction properties for products of Markov driven random matrices. J. Math. Phys. Anal. Geom. 4, 457–489 (2008)zbMATHMathSciNetGoogle Scholar
  27. GL.
    Guivarc’h Y., Le Jan Y.: Winding of the geodesic flow on modular surfaces. Ann. Sci. Ec. Norm. 26, 23–50 (1993)zbMATHMathSciNetGoogle Scholar
  28. He.
    Hejhal, D.A.: The Selberg Trace Formula for PSL(2, \({\mathbb{R}}\)), vols. I, II. Lecture Notes in Mathematics, vols. 548, 1001. Springer, BerlinGoogle Scholar
  29. Hö.
    Hörmander L.: Notions of Convexity. Progress in Mathematics, vol. 127. Birkhäuser, Boston (1994)Google Scholar
  30. I.
    Iwaniec, H.: Spectral methods of automorphic forms. Second edition. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence; Revista Matemética Iberoamericana, Madrid (2002)Google Scholar
  31. Kai.
    Kaimanovich V.: Discretization of Bounded Harmonic Functions on Riemannian Manifolds and Entropy. Potential Theory (Nagoya, 1990), pp. 213–223. de Gruyter, Berlin (1992)Google Scholar
  32. Kal.
    Kalinin B.: Livšic theorem for matrix cocycles. Ann. Math. 173, 1025–1042 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  33. Kap.
    Kapovich M.: Hyperbolic Manifolds and Discrete Groups. Progress in Mathematics, vol. 183. Birkhéuser, Boston (2001)Google Scholar
  34. KM.
    Kleinbock D., Margulis G.A.: Logarithm laws for flows on homogeneous spaces. Invent. Math. 138, 451–494 (1999)zbMATHMathSciNetCrossRefADSGoogle Scholar
  35. Le1.
    Le Page, É.: Théorèmes limites pour les produits de matrices aléatoires. In: Heyer, H. (ed.) Probability Measures on Groups. Lecture Notes in Mathematics, vol. 928, pp. 258–303 (1982)Google Scholar
  36. Le2.
    Le Page : Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Stat. 25(2), 109–142 (1989)zbMATHMathSciNetGoogle Scholar
  37. Lu.
    Lubotzky A., Meiri C.: Sieve methods in group theory I: powers in linear groups. J. Am. Math. Soc. 25, 1119–1148 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  38. LS.
    Lyons T., Sullivan D.: Function theory, random paths and covering spaces. J. Differ. Geom. 19(2), 299–323 (1984)zbMATHMathSciNetGoogle Scholar
  39. M.
    Margulis G.A.: On Some Aspects of the Theory of Anosov Systems. With a Survey by Richard Sharp. Springer Monographs in Mathematics. Springer, Berlin (2004)CrossRefGoogle Scholar
  40. Su1.
    Sullivan D.: Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149, 215–237 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  41. Su2.
    Sullivan D.: Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups. Acta Math. 155(3-4), 243–260 (1985)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CNRS, DMA, École Normale SupérieureParisFrance
  2. 2.LAMA, Université Paris-Est Marne-la-ValléeChamps sur MarneFrance

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