New York Journal of Mathematics
Volume 21 (2015) 511-531


Vaibhav Gadre, Joseph Maher, and Giulio Tiozzo

Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

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Published: July 2, 2015
Keywords: Fuchsian group, Lyapunov exponent, word metric, random walk
Subject: 37C85, (37A50, 37D25, 60B15)

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.


The first author was supported by a Global Research Fellowship with the Institute of Advanced Study at the University of Warwick. The second author would like to thank Kathi Crow for her generous hospitality, and was supported by PSC-CUNY award 44-178 and Simons Foundation grant CGM 234477.

Author information

Vaibhav Gadre:
University of Warwick

Joseph Maher:
CUNY College of Staten Island and CUNY Graduate Center

Giulio Tiozzo:
Yale University