Research in Combinatorial Group Theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The investigator will work on further development and applications of his powerful geometric machinery of graded diagrams which was created to solve the famous Burnside problem (posed in 1902) on periodic groups of sufficiently large even exponents. In particular, he will extend the framework of the main induction construction of this machinery to obtain more results on groups whose defining relators are nth powers. In addition, he will work on a generalized version of the Hanna Neumann conjecture on subgroups of free groups and long-standing Stallings' problem on presentations of the trivial group. This research is in combinatorial group theory which studies groups defined by means of generators and defining relators and is at the intersection of the theory of groups with low dimensional topology, geometry, and mathematical logic. The theory of groups is a mathematical theory of symmetry which interacts with many other disciplines, for example, physics and chemistry outside of mathematics, coding theory, number theory, topology and geometry inside mathematics.