Michelle Bucher-Karlsson: Exponential growth of group

Let G be a group generated by a finite set S. Let b(n) be the number of elements in G that can be written as a product of n or less elements from S and S^{-1}. We will see that the growth function b(n) depends on the generating set S, but not too much. It can be of three different types: polynomial, exponential, or intermediate. Giving a geometric way of attributing a number theoretical function to any (finitely generated) group, it has been studied extensively, culminating with the determination of groups of polynomial growth by Gromov, and the construction of intermediate groups by Grigorchuk. 

Finding free subgroups or semigroups is a classical way of ensuring exponential growth. In this talk, I will describe how this can be done for groups acting on trees, leading to answers of questions of Avinoam Mann on the minimal exponential growth rates of free products, amalgamated products and HNN extensions. This is joint work with Alexey Talambutsa.