15:30 - 16:00 (teatime in the lounge)

16:00 - 17:00 (lectures in Lecture Room S-1)

Speaker:Laurent Bartholdi (Göttingen University)

Title:Growth and Poisson boundaries of groups

Abstract:

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.