Luke Jeffreys

The statistical hyperbolicity of Teichmüller space

Introduced by Duchin-Lelièvre-Mooney, the notion of statistical hyperbolicity encapsulates whether a space is ‘on average’ hyperbolic at large scales. That is, a metric space is said to be statistically hyperbolic if the average distance between pairs of points on large spheres, divided by the radius, converges to two as the radius tends to infinity. Gromov-hyperbolicity of a metric space is neither necessary nor sufficient for statistical hyperbolicity. In joint work with Aitor Azemar and Vaibhav Gadre, we show that Teichmüller space, in general not Gromov-hyperbolic, is statistically hyperbolic for a large class of measures including the Lebesgue class measures for which statistical hyperbolicity was already established by Dowdall-Duchin-Masur.