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Lyapunov Exponents for Surface Group Representations

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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.

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Authors and Affiliations

  1. CNRS, DMA, École Normale Supérieure, 45 rue d’Ulm, 75005, Paris, France

    Bertrand Deroin

  2. LAMA, Université Paris-Est Marne-la-Vallée, 5 boulevard Descartes, 77454, Champs sur Marne, France

    Romain Dujardin

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  1. Bertrand Deroin
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  2. Romain Dujardin
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Correspondence to Romain Dujardin.

Additional information

Communicated by S. Zelditch

B.D.’s research was partially supported by ANR-08-JCJC-0130-01, ANR-09-BLAN-0116. Both authors partially supported by ANR project LAMBDA, ANR-13-BS01-0002.

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Deroin, B., Dujardin, R. Lyapunov Exponents for Surface Group Representations. Commun. Math. Phys. 340, 433–469 (2015). https://doi.org/10.1007/s00220-015-2469-7

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