Geometry of Artin Group Actions

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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.