Geometric Group Theory and Rewriting Systems

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

Abstract for Hermiller's proposal Geometric group theory and rewriting systems This project involves three main areas of exploration in the field of geometric group theory, using both computational techniques (in particular, rewriting systems) and geometric methods to study groups. The first area is the extension and application of geometric group theory techniques to groups, monoids, and algebras. The second topic is a study of polyfree groups, including algorithmic problems for these groups and connections to braid groups and Artin groups. The third area is a study of well-founded and admissible orderings, including constructions of rewriting systems which have applications to noncommutative Groebner bases. Groups are a useful mathematical tool which originated in the study of symmetry, including symmetries of naturally occurring objects such as crystals and molecules. For example, the reflections and rotations of the plane that map a square back to itself form a group. This project is in a field which is at the interface between group theory, geometry, and computer science. The principal investigator's work focuses on the interplay between these areas, and in particular on the pursuit of computationally effective and efficient algorithms for working with groups that are associated with certain geometric structures. This project also includes applications of these techniques to other areas of algebra.