Geodesic ray tracking for random walks on groups

Giulio Tiozzo (Harvard)

Abstract: Given a finitely generated group G acting on a geodesic space X and a probability measure on G, one can construct a random walk by choosing at each step a random group element and making it act on X. The natural question arises whether the sample paths can be approximated by some geodesic in X. We will prove that, in a quite general setting, the sample path and the limiting geodesic lie within sublinear distance. Our argument includes the case of the mapping class group acting on Teichmueller space, answering a question of Kaimanovich. Another application includes the statistics of excursions of random Teichmueller geodesics in the thin part of moduli space.