Vanishing of the Lyapunov expansion exponent and word length statistics along random geodesics

For a finitely generated group of circle diffeomorphisms, Deroin-Kleptsyn-Navas defined a Lyapunov expansion exponent at a point on the circle. They conjectured that for non-uniform lattices in SL(2.R), the exponent vanishes at Lebesgue almost every point on the circle. In
joint work with J. Maher and G. Tiozzo, we prove this conjecture by studying the statistics of the growth of word length along random geodesics. If time permits, I will explain the analogous statistical results along random Teichmuller geodesics.