Research in Geometric Group theory: Artin groups and Automorphism Groups

Brandeis University, Waltham MA

Investigators

Abstract

This project has two main parts. The first concerns automorphism groups of right-angled Artin groups. Right-angled Artin groups are finitely presented groups whose only relators are commutators. They may be viewed as interpolating between free groups and free abelian groups. Thus, their (outer) automorphism groups interpolate between Out(F_n) and GL(n,Z) and provide a context for understanding the similarities and differences between these groups. The goal is to generalize techniques used in the study of automorphism groups of free groups---such as the construction of a contractible ""outer space"" with a proper Out(F_n) action - and use these techniques to study properties of automorphism groups of arbitrary right-angled Artin groups. The second part of the project concerns more general Artin groups associated to infinite Coxeter groups. The project will address a number of questions regarding the coarse geometry of these groups. Geometric group theory may be viewed as the study of symmetry groups of geometric objects. A particularly rich class of geometric objects are cubical complexes with nice local structure. These complexes serve as models for a variety of problems in robotics and have applications to other areas of mathematics. Right-angled Artin groups arise as groups of symmetries of certain cubical complexes. Moreover, the groups themselves have interesting symmetries known as automorphisms. This project will investigate properties of these groups, their automorphisms and the associated geometries. In addition to right-angled Artin groups, the project will study the large scale geometry of more general Artin groups. Some Artin groups, such as braid groups, are well understood and have important applications to topology, cryptography, and other fields. Other Artin groups are much more mysterious and some of the most fundamental problems in group theory, such as the ""word"" and ""conjugacy"" problems, remain unanswered for these groups. By studying their large scale geometry, the project aims to gain insight into some of these more mysterious groups.