Abstract
In this paper we explore the idea that Teichmuller 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 Teichmuller space. We consider several different measures on Teichmuller 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 Teichmuller space, showing that “most triangles are mostly thin.”
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