The main aspect of the project is to study the expansion properties of Cayley graphs and to understand various representation theoretic properties of pro-finite and discrete groups. More specifically, the PI will study the representation theory of the automorphism groups of free groups and the universal lattices; and will use the results to construct new examples of expander graphs. This may lead to better understanding widely used algorithms in computational group theory like the product replacement algorithm. Furthermore, findings from this project may have connections to more distant topics like three-dimensional manifolds via the Thurston Conjecture and the Virtual Haken Conjecture.
Expanders are highly connected finite graphs which have many applications in combinatorics and theoretical computer science. Informally, a graph is an expander if it cannot be separated into two large pieces by removing a small number of vertices and the adjoining edges. A standard counting argument shows that a randomly chosen graph is a good expander. Unfortunately this argument does not allow for the explicit construction of expanders, required for many applications. The first explicit expander graphs were constructed by Margulis using Kazhdan property T. The aim of this project is to understand how good these expanders are by estimating the related Kazdan constants. As well as objectives which would be both deep and of practical use, the project has elementary aspects which can be presented at an undergraduate level and thus serve educational purposes.
