Geometry of Artin Group Actions

Groups are a structure that encode the mathematical idea of symmetry. While one may think of symmetry as a geometric notion, such as in the symmetries of a snowflake or of a wall-paper, mathematicians investigate a more abstract notion of symmetry, namely the possible transformations that leave features of an object unchanged. This larger class of symmetries ranges from shuffling a deck of cards to the manipulation of strings of hair when making a braid. The study of this generalised idea of symmetry, known as group theory, plays a key role in modern mathematics, as understanding the symmetries of an object is a stepping stone towards a deeper understanding of that object.

Geometric group theory is the field of mathematics that aims to understand these more abstract symmetry groups by realising them as symmetries of new geometric objects. Doing so allows one to use geometric methods to investigate the structure of these groups, and the resulting dialogue between algebra and geometry has proved particularly fruitful in recent years, both within and outside mathematics.

This project focuses on a class of groups known as Artin groups, a vast generalisation of the groups involved in making braids, which have ramifications in many areas of mathematics and beyond. While the structure of braid groups is relatively well understood, the situation is much more mysterious for general Artin groups, and many important and natural questions remain open.

This project will introduce a new geometric framework to study general Artin groups. In recent years, large classes of groups from various horizons have been studied with great success from a geometric viewpoint, and particularly from the point of view of actions on spaces satisfying some form of non-positive curvature. This is such an approach that will be carried out in this project. More precisely, this project will study large classes of Artin groups through their actions on hyperbolic spaces, and will use the dynamics of such actions to understand the structure of these groups in great generality. This project will also highlight structural similarities with other important classes of groups.

This work represents an exciting project at the crossroads between algebra, combinatorial geometry, and dynamics in negative curvature. It will involve collaborations with researchers from Canada and France. A workshop will be organised halfway through, in order to bring together experts studying Artin groups from various perspectives: algorithmic group theory, combinatorics, etc.

The goal of this project is to bring new insights in the study of Artin groups, and to highlight similarities with other important classes of groups, such as mapping class groups and cubical groups. The main impact of this work will thus be on academics working in geometric group theory and in its neighbouring fields: algorithmic group theory, combinatorics, etc. Researchers will be able to use this new point of view to study Artin groups, leading to further research on this topic. Moreover, as the class of groups studied from the point of view of actions on hyperbolic spaces is an ever-growing on, we expect this work to have significant impact in geometric group theory, as researchers will be able to import the techniques introduced in this project to study other classes of groups.

A central part of this project is the organisation of a workshop in Spring 2020 that will bring together experts from various horizons. Indeed, Artin groups are a topic of active research at the intersection of several fields (hyperbolic geometry, Garside structures, etc. ), but where communication barriers may impede the cross-fertilisation of these fields. By bringing together leading experts working on these groups from various perspectives, we expect a significant transfer of knowledge between these various subfields. This workshop will also spark new collaborations, leading to new and exciting approaches to study Artin groups and related groups.

Concepts and tools at the heart of this project have already found applications outside of mathematics: CAT(0) cube complexes have been used to model configuration spaces relevant to robotics, while small cancellation techniques have strong connections with communication networks. Artin groups represent a topic that is particularly easy to introduce, and the PI will thus propose the (co-)organisation of specialised courses, both at Heriot-Watt University and through the Scottish Mathematical Sciences Training Centre, thus allowing large cohorts of students across Scotland to learn about cutting-edge research in geometric group theory. We expect a long-term economic impact since, as often with pure mathematical research, providing students with today's abstract mathematical frameworks will enable them to model and solve tomorrow's applied problems.