Groups are mathematical objects that can be understood as an abstract model for symmetry. While the concept has been around since the 19th century, group theory is still a very active research field. This is due to the fact that symmetry plays a central role not only in mathematics, but also in natural sciences such as physics and chemistry as well as in computer science.
In geometric group theory, we don't investigate abstract group properties directly, but look at Cayley graphs instead. Cayley graphs are geometric objects which encode information about the structure of the group. In general there are many different Cayley graphs for a group. A geometric group property in this context is a property of those Cayley graphs which only depends on the underlying group and not on the specific choice of the graph.
Some particularly interesting geometric group properties are closely related to random processes taking place on those graphs. Originally motivated by physical applications the study of such processes has now taken a mathematical life of its own and has become an active and fast growing branch of mathematics.
The project „Graphs and groups“ aims to contribute to our understanding of random processes on Cayley graphs. This will be achieved by two complementary lines of research:
The first part of the project continues ongoing research on the structure of so called graph wreath products of groups. We are interested in those products mainly because they can potentially be utilised to construct groups with unexpected properties. Among other things this could lead to a counterexample to a conjectured connection between the cost and l2-Betti numbers of groups.
The second part of the project is concerned with substructures of Cayley graphs that could greatly simplify studying random processes on Cayley graphs. Specifically we aim to find certain suitably embedded substructures (grids and trees) of Cayley graphs. Since random processes are rather well understood on grids and trees this will lead to a significantly better understanding of random processes on more general Cayley graphs.