Abstract
We give a survey of recent results on the Poisson-Furstenberg boundaries of random walks on groups, and their applications. We describe sufficient conditions for random walk to have non-trivial boundary, or, on the contrary, to have trivial boundary. We review recent progress in description of the boundary for random walks on various groups, including wreath products. We describe how the Poisson-Furstenberg boundary can be used to obtain lower bounds for the growth function of the groups of intermediate growth. We also discuss relation between properties of the boundary with other asymptotic properties of groups, including isoperimetry and various characteristics of random walks.
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