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

Abstract

For a non-uniform lattice in SL$(2,\mathbb{R})$, we consider excursions of a random geodesic in cusp neighborhoods of the quotient finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmüller setting, we consider invariant measures for the SL$(2,\mathbb{R})$ action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmüller geodesic random with respect to a SL$(2,\mathbb{R})$-invariant measure, we study its excursions in thin parts of the associated affine invariant submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel–Veech theory, these are given by various Siegel–Veech constants. As a direct consequence, we show that the word metric of mapping classes that approximate a Teichmüller geodesic ray that is random with respect to the Masur–Veech measure, grows faster than $T$ log $T$.