Lyapunov exponents for surface groups representations
arXiv:1305.0049
Abstract
Let (ρ_\la)_{\la\in \La} be a holomorphic family of representations of a surface group π_1(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 \La, that is a (1,1) positive closed current attached to the bifurcations of the family. It is defined as the of the Lyapunov exponent of the representation with respect to the Brownian motion on the Riemann surface S, endowed with its Poincare metric. We show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in \La. For instance, the random hypersurfaces of \La defined by the condition that a random closed geodesic on S is mapped under ρ_\la to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current. The proofs are based on our previous work "Random walks, Kleinian groups, and bifurcation currents", and on a careful control of a discretization procedure of the Brownian motion.
Title slightly changed. We changed the normalization for hyperbolic Brownian motion, whose generator is now the Laplacian (instead of half of it). As a consequence, several formulas in the paper are changed by a factor 2