seminar: Cusp excursions of Teichmüller geodesics and random walks on the mapping class group
speaker: Giulio Tiozzo (Yale University)abstract: Given a finitely generated group G acting on a space X and a probability measure on some generating set, one can consider the random walk determined by choosing at each step a random generator according to the given probability measure. The random walk determines a harmonic measure on a suitable boundary of X, namely the probability that the walk hits a given subset of the boundary. On the other hand, the boundary might possess a natural measure of geometric origin, for instance a Lebesgue measure, hence it is interesting to ask whether the harmonic measure for some walk coincides with the Lebesgue measure. We will consider the case when the group is the mapping class group MCG of isotopy classes of diffeomorphisms of a surface, which acts naturally on Teichmueller space. By measuring different statistics along typical geodesics for both measures, we will prove that the harmonic measure is singular with respect to Lebesgue measure. We will also discuss new results about approximation of sample paths with Teichmueller geodesics. This is joint work with Vaibhav Gadre and Joseph Maher.
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