Laurent Bartholdi: Growth and Poisson boundaries of groups

Let G be a finitely generated group. A rich interplay between algebra and geometry arises by viewing G as a metric space, or as a metric measured space. I will describe two invariants of finitely generated groups, namely growth and Poisson boundary, and explain by new examples that their relationship is deep, but still mysterious. Its growth function counts the number of group elements that can be written as a product of at most n generators. This function depends on the choice of generators, but only mildly. For example, the growth of Z^d is asymptotic to n^d, while the growth of a free group is asymptotic to 2^n. There are groups whose growth function is known to lie strictly between polynomials and exponentials; I will describe the first examples for which the asymptotic growth is known. I will also describe an example of a group of exponential growth, whose Poisson boundary is trivial for all finitely-supported random walks. Perhaps surprisingly, both examples come from the same general construction, permutational wreath products.

This is joint work with Anna Erschler.