Abstract
Abstract We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moments with respect to the Teichmüller metric and whose supports generate nonelementary subgroups. We prove that Teichmüller space with the Teichmüller metric is statistically hyperbolic for such a harmonic measure.
Highlights
1 Introduction The notion of statistical hyperbolicity, introduced by Duchin–Lelièvre–Mooney [4], encapsulates whether a space is on average hyperbolic at large scales, that is, for any point in the space and spheres centred at that point whether as the radius r → ∞, the average distance between pairs of points on the sphere of radius r is 2r
For many Lebesgue-class measures on Teichmüller space, Dowdall–Duchin– Masur showed that Teichmüller space with the Teichmüller metric is statistically hyperbolic
We consider the same question for harmonic measures that arise from random walks on the mapping class group determined by probability distributions with finite 1st moment with respect to the Teichmüller metric and whose supports generate nonelementary subgroups
Summary
The notion of statistical hyperbolicity, introduced by Duchin–Lelièvre–Mooney [4], encapsulates whether a space is on average hyperbolic at large scales, that is, for any point in the space and spheres centred at that point whether as the radius r → ∞, the average distance between pairs of points on the sphere of radius r is 2r. We consider the same question for harmonic measures that arise from random walks on the mapping class group determined by probability distributions with finite 1st moment with respect to the Teichmüller metric and whose supports generate nonelementary subgroups. Let μ be a probability distribution on the mapping class group Mod(S) with finite 1st moment with respect to the Teichmüller metric, and such that the support generates a nonelementary subgroup. When μ has finite 1st moment in the word metric, by a theorem of Guivarch–LeJan [10], the harmonic measure from the μ-random walk is singular with respect to the Lebesgue measure class. We note that when n > 2 and is a nonuniform lattice, Randecker–Tiozzo proved that a harmonic measure arising from a μ whose support generates and has finite (n−1)th moment with respect to a word metric is singular. We assume that the complex dimension of T (S) is greater than one and present Theorem 1.1 with that assumption
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