Research Council - Backup mandate Doryan Temmerman - Dichotomies of CAT(0) groups and integral group rings with their relation to geometrical structures.

Symmetry is everywhere around us in day-to-day life. From the shape of a football to the layout of the tiles in your bathroom. In mathematics, we study symmetries via objects which are called groups, in the domain aptly named "Group Theory". At least, this is the way symmetries were studied historically. In modern days, groups are often defined and studied without an explicit symmetry
associated to them.
For these so called abstract groups, there is a whole domain of mathematics (called "Representation Theory") dedicated to retrieving the symmetries which belong to it. This is done by studying on which higher dimensional space such a group defines a symmetry and what this symmetry is exactly.
One way of doing this for a group G is by studying a special group which can be associated to G, which we will denote by U(ZG). In this project, I will study this group U(ZG) in detail. More specifically, I will ask the question whether or not it has a symmetry associated to it that resembles a "tree" or other geometric structure. Moreover, if the answer is yes, what extra special properties of this symmetry can be expected? Can it have fixed points? And what does this tell us
exactly about the structure of U(ZG) as an abstract group?
One of the geometric objects that I will study more in detail are the so called CAT(0) spaces. These are spaces which are deformed so that the triangles in this space look "slimmer" than the triangles in our real world