数理談話会

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 γ(n) 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.
I will show that, for almost any function γ that grows sufficiently fast, there exists a group with growth asymptotic to γ. These give also the first examples of groups for which the growth function is known, and is neither polynomial nor exponential.
The Poisson boundary of a random walk (given by a one-step measure) describes the tail events of the walk. If the random walk takes place on a group G, then the Poisson boundary is very much connected to classical invariants of G: if the boundary is trivial, then the group generated by the measure's support is amenable. If the boundary is non-trivial for a finitely supported measure, then G has exponential growth.
The connection between growth and Poisson boundary, however, is still mysterious. Kaimanovich and Vershik asked in 1983 whether the converse holds; namely, whether there exist groups of exponential growth such that all measure with finite support have trivial boundary.
I will show that such examples exist. Curiously, the constructions of all examples are based on a common method, that of "permutational wreath products". I will outline a few other consequences of the construction to geometric group theory.
This is joint work with Anna Erschler.