Lyapunov Exponents for Surface Group Representations

Let (ρλ)λ∈Λ be a holomorphic family of representations of a surface group π1(S) into PSL(2,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.