The focal area of the program lies at the juncture of three areas: Probability theory of random walks,Ergodic theory of flows on negatively curved spaces,Gromov hyperbolic groups.Random walks on finite and infinite, finitely generated groups is a topic of considerable vintage. It is well-known, starting with Kesten's characterization of amenability, that asymptotic properties of such random walks are intimately connected to the large scale geometry of the underlying group. Vershik and Kaimanovich (1983) introduced entropic techniques to study the Poisson boundary of random walks on countable discrete groups, which is a natural measure theoretic space "at infinity" associated with the random walk. In a seminal paper in 2000, Kaimanovich gave a very general sufficient condition on the one step distribution of the walk on a hyperbolic group for the Poisson boundary to equal the (geometric) Gromov boundary. Further probabilistic methods have started being applied to hyperbolic groups relat...
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