This project involves problems in geometric group theory and spaces of non-positive curvature with a focus on Artin groups and Garside groups. Artin groups span a wide range of groups including braid groups, free groups, and free abelian groups. The first part of the project concerns automorphism groups of right-angled Artin groups. These are finitely generated groups whose presentations involve only commutator relations between generators. By varying the number of commutator relations, right-angled Artin groups may be viewed as "interpolating" between free groups and free abelian groups. The automorphism groups of free groups have been the focus of much research in recent years. They have been shown to have much in common with linear groups, in particular with the automorphism groups of free abelian groups. This project aims to put these results in a broader context by studying the automorphism group of a general right-angled Artin group. The key idea is to construct a contractible space, analogous to Culler and Vogtmann's "outer space", on which the automorphism group acts.
The second part of the project concerns Garside groups. Garside groups are groups with algorithmic properties similar to those of braid groups. A Garside structure on a group gives a very powerful tool for studying the combinatorial and geometric properties of the group. This approach has been used extensively in the study of finite type Artin groups. While no such structure exists for infinite type Artin groups, it appears that something close to a Garside structure may exist for at least some of these groups. The project will consider various generalizations of Garside groups and their properties.
Symmetries of geometric objects have been studied since ancient times and have played an important role in many areas of mathematics and the sciences. They form the model for the abstract mathematical notion of a "group". Groups arise in nearly every field of mathematics. While not all groups occur naturally as symmetry groups, it is always possible to construct geometric objects on which a given group acts as symmetries. In recent years, this interplay between groups and geometry has been increasingly exploited to understand various types of infinite groups. A particularly interesting class of groups that lends itself to such techniques are the "braid groups". The n-strand braid group, as the name suggests, encodes the different ways that a set of n strings can be braided. It also describes the different ways that a collection of n particles can move around in a plane. Braid groups have applications to topology, mathematical physics, and cryptography. Another fundamental class of groups are the "free groups". These groups form the foundation for studying algorithmic and combinatorial properties of groups in general. In this project we will study a large class of groups, known as Artin groups, that includes the braid groups, the free groups, and many others. The project aims to reach a better understanding of Artin groups by studying geometric objects associated with them, as well as searching for new algorithmic techniques.
