Energy, product sets and growth in infinite groups

The purpose of the Erwin Schrödinger project Energy, product sets and growth in infinite groups is the study of abstract algebraic objects, so called groups. However, often they are much less abstract than it might seem. Many groups do indeed occur in a quite natural way as symmetries of geometric spaces: the well- known planar Euclidean plane, the negatively curved hyperbolic plane, or highly connected networks, for example. Hence, a good way to approach groups is to understand their geometry. In this project, I will work with hyperbolic groups. In these groups much negative curvature behaviour is preserved. In order to analyse the complexity of groups and to distinguish them from each other, we often consider their growth rate. In hyperbolic groups this growth rate is exponential, i.e. like the growth rate of bacteria whose population doubles in every reproduction step. A new trend is the study of product set growth in groups. We recently developed new geometric tools to understand this in hyperbolic groups. In this project I will address this question in other interesting groups, that is, in infinite finitely generated groups each element of which has a uniform finite order (for example, every power of thousand equals to 1). The construction of these so called Burnside groups is one of the highlights of my field and is quite important to test new hypotheses or methods. On the other hand, rather recent geometric methods allow for approximations of these groups by the well- studied hyperbolic groups. I want to make use of this approximation to transfer our recently acquired knowledge on hyperbolic groups to Burnside groups. I expect to improve our understanding of the growth of product sets in Burnside groups, their complexity and the quality of the aforementioned approximations. The Erwin Schrödinger Fellowship allows me to conduct this project in collaboration with a leading expert in the geometric construction of Burnside groups and their growth at the University of Rennes in France.

We proved fine and optimal estimates on the growth of product sets, or sub-semi-groups respectively, in Burnside groups of sufficiently large odd exponent. With these results, we extended previously known growth estimates of Razborov and of Safin for free groups, as well as of Delzant and PI for hyperbolic groups, to Burnside groups. For this purpose, we have, in particular, further developed methods to study product set growth in groups that act acylindrically on a hyperbolic space.