Word length statistics for Teichmüller geodesics and singularity of harmonic measure

Abstract

Given a measure on the Thurston boundary of Teichmüller space, one can pick a geodesic ray joining some basepoint to a randomly chosen point on the boundary. Different choices of measures may yield typical geodesics with different geometric properties. In particular, we consider two families of measures: the ones which belong to the Lebesgue or visual measure class, and harmonic measures for random walks on the mapping class group generated by a distribution with finite first moment in the word metric. We consider the ratio between the word metric and the relative metric of approximating mapping class group elements along a geodesic ray, and prove that this ratio tends to infinity along almost all geodesics with respect to Lebesgue measure, while the limit is finite along almost all geodesics with respect to harmonic measure. As a corollary, we establish singularity of harmonic measure. We show the same result for cofinite volume Fuchsian groups with cusps. As an application, we answer a question of Deroin-Kleptsyn-Navas about the vanishing of the Lyapunov expansion exponent.

Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials

“…It was shown using [15,Proposition 5.4] that along Leb-typical geodesic rays E(γ, T)/T → ∞. We prove here: 1].…”

“…The projected path p(x 0 , γ T ) is a quasi-geodesic in ( X thick , d thick ) [15,Lemma 5.1]. More precisely, L(x 0 , γ T ) − d thick (x 0 , γ T ) grows at most linearly in N. As we show in Lemma 3.8, N grows linearly in T. Hence, we get Theorem 1.6.…”

“…Because of the compactness of the thick part, this diameter is finite. As shown in [15,Proposition 3.11], along a recurrent Teichmüller geodesic γ the total excursion E(γ, T) in the Masur collection gives a coarse lower bound on the word metric of the approximating group elements h T , i.e. there exists constants a 1 , a 2 > 0 such that…”

“…Acknowledgements. The central question considered in the paper arose in joint work with J. Maher and G. Tiozzo [15]. I thank them for the initial discussion and for brining the paper by Diamond and Vaaler [9] to my attention.…”

“…In fact, if the contribution from the largest excursion is removed, then the word metric grows like T log T up to uniform multiplicative and additive constants. Theorem 1.7 should be thought of as a refinement of Proposition 5.6 in [15] which states that along a Leb-generic geodesic ray the ratio d G (1, h T )/T goes to infinity as T → ∞.…”