Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the boundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give an affirmative answer to this question, by considering the behavior of the word metric along typical geodesic rays and their excursions into cusps. We also consider the behavior of the word metric along rays chosen according to harmonic measure on the boundary, arising from random walks with finite first moment. We show that the excursions have different behavior in the Lebesgue measure and harmonic measure cases, which implies that these two measures are mutually singular.