This project is primarily concerned with Golod-Shafarevich groups, a class of groups which admit a presentation with a small set of relations. Golod-Shafarevich groups have been originally introduced as a tool for solving two outstanding problems -- the class field tower problem and the general Burnside problem -- and continue to play an important role in various areas of mathematics, including geometric group theory, algebraic number theory and three-manifold topology. The Principal Investigator will continue studying the subgroup structure of these groups, their asymptotic invariants (e.g., rank gradient and subgroup growth) and certain aspects of their representation theory (primarily properties (T) and (tau)). Special attention will be devoted to Golod-Shafarevich groups of number-theoretic origin, e.g. Galois groups of pro-p extensions of number fields with restricted ramification, and Kac-Moody pro-p groups over finite fields. The Principal Investigator will also continue his work on constructing new groups with property (T) with prescribed largeness properties.
Groups play a fundamental role in mathematics by describing symmetries of various objects like geometric figures or number systems. A group can often be presented by generators and relations which provide a simple way to define the group but usually offer little insight into its structure. This project will develop new tools that can help better understand a group based on its presentation by generators and relations which, in turn, can yield new information about the object whose symmetries the group describes. The findings of the project will likely have applications beyond group theory, e.g., in the areas of three-manifold topology, number theory and graph theory.
