Geometric group theory explores the interaction between groups and geometry. Of central interest are geometries satisfying certain curvature conditions and the groups that act on them. This project concerns right-angled Artin groups, a class of groups that interpolates between free groups and free abelian groups. These groups have played an increasingly important role in geometric group theory and low dimensional topology in recent years. The action of right-angled Artin groups on cubical complexes satisfying a non- positive curvature condition, known as the CAT(0) condition, has been central to our understanding of these groups as well as to many of their applications. Moreover, these actions serve as a rich class of examples for understanding more general CAT(0) spaces. This project is designed to further our understanding of right-angled Artin groups, their automorphisms, and their action on cubical complexes, and to explore implications for more general CAT(0) spaces.
Geometric group theory is a young and exciting field that combines techniques from several areas of mathematics to give new insight into groups of symmetries of geometric objects. Symmetries play an important role throughout the sciences, particularly in chemistry, physics, and astronomy. The goal of this project is to develop a deeper understanding of symmetry groups of a certain type of geometric object, known as a cubical complex. Cubical complexes arise in many contexts. For example, they can be used to model robotic systems, so that mathematical properties of the geometry translate directly into physical properties of the robot. This is a growing area of research with potential for many applications. In addition, the PI is involved in a variety of activities designed to engage young people, particularly women, in mathematics. Ideas from this project are being used to involve these young people in the frontier of mathematics.