Statistical Hyperbolicity for Harmonic Measure

Statistical hyperbolicity in groups

In this paper, we introduce a geometric statistic called the sprawl of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to ...

Statistical Hyperbolicity in Teichmüller Space

In this paper we explore the idea that Teichmüller space with the Teichmüller metric is hyperbolic “on average.” We consider several different measures on Teichmüller space and show that with respect to each one, the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible.

Brownian Motion, Harmonic Functions and Hyperbolicity for Euclidean Complexes

Rigidity of harmonic measure

Let J be the Julia set of a conformal dynamics f . Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out...

Recurrence of Quadratic Differentials for Harmonic Measure

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is...