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

Gadre, V. (2017) Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials. Journal of the European Mathematical Society, 19(10), pp. 3053-3089. (doi: 10.4171/JEMS/735)

Abstract

For a non-uniform lattice in SL(2, 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 Teichmuller setting, we consider invariant measures for the SL(2, 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 Teichmuller geodesic random with respect to a SL(2,R)-invariant measure, we study its excursions in thin parts of the associated 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 Siegel-Veech constants. As a direct consequence, we show that the word metric of mapping classes that approximate a Teichmuller geodesic ray that is random with respect to the Masur-Veech measure, grows faster than T log T.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gadre, Dr Vaibhav
• Authors: Gadre, V.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of the European Mathematical Society
• Publisher: European Mathematical Society
• ISSN: 1435-9855
• ISSN (Online): 1435-9863
• Copyright Holders: Copyright © 2017 European Mathematical Society
• First Published: First published in Journal of the European Mathematical Society 19(10):3053-3089
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher