The University of Melbourne Growth of Finitely Generated Groups

To every finitely generated group G we can assign an equivalence class of growth function. That is, a function describing for each natural number k the number of elements that can be reached in the Cayley graph of G within k steps. By calculating such a function it becomes possible classify groups into three categories of growth: polynomial, intermediate and exponential. We will look at what properties of a group can be deduced by its growth type, in particular proving Gromov’s theorem on polynomial growth, and introduce the notions of growth tightness and uniform exponential growth in order to bound the growth of various groups. We will then finish by looking at the currently unsolved problem of the growth of Thompson’s group F