Statistical Hyperbolicity in Teichmüller Space

Luke Jeffreys

Friday 28th April, 2017 16:00-17:00 ?

Abstract

Hyperbolic spaces and spaces of non-positive curvature are important objects of study throughout many areas of mathematics. Not only do they have many useful properties that assist in the study of related objects, but they also appear almost ubiquitously. For example, almost every surface is hyperbolic; almost every group is hyperbolic, in a reasonable sense; and, heuristically speaking, a generic 3-manifold is hyperbolic. It is therefore a worthwhile endeavour to investigate the hyperbolic properties of a space.

A tool used for this purpose, and introduced by Duchin-Lelièvre-Mooney, is the notion of sprawl for a metric space. The sprawl of a metric space is related to the average distance between points in spheres and a space having sprawl equal to two is said to be statistically hyperbolic. What is particularly interesting about this notion is that spaces which we would have initially assumed would be statistically hyperbolic are not necessarily so and vice versa. That is, the sprawl of a space is a more sensitive measure of the large-scale curvature.

In this talk, we will introduce the necessary notions to properly define the sprawl of a metric space and investigate this property for some interesting examples. We will also introduce and discuss the phenomenon of statistical hyperbolicity in Teichmüller space, an important object of study in its own right, and consider why such an investigation is interesting.

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