Statistical hyperbolicity in Teichm\"uller space
Abstract
In this paper we explore the idea that Teichm\"uller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichm\"uller space. We consider several different measures on Teichm\"uller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichm\"uller space, showing that "most triangles are mostly thin."
Cite
@article{arxiv.1108.5416,
title = {Statistical hyperbolicity in Teichm\"uller space},
author = {Spencer Dowdall and Moon Duchin and Howard Masur},
journal= {arXiv preprint arXiv:1108.5416},
year = {2013}
}
Comments
v5: 41 pages. Added a new theorem quantifying the "expected thinness" of random geodesic triangles Teichmuller space, as well as an appendix that gives a self-contained account of some relevant facts about random walks in Teichmuller space. The treatment of "reverse triangle inequalities" has also been adjusted. This version accepted for publication in the journal Geometric And Functional Analysis