The Geometry of the Isomorphism Problem for Artin Groups

The notion of group is a mathematical concept that finds its roots in the symmetries of geometric shapes, but which is now used more generally to model reversible transformations of abstract systems, such as the shuffling of cards and the braiding of strands. Geometric group theory is the field of mathematics that studies groups by realising them as symmetries of complex geometric objects. It is thus a field that uses geometric tools to answer questions coming from algebra. It finds itself at the interface between algebra, geometry, topology, and combinatorics, and has found applications in fields such as robotics and cryptography. This project focuses on Artin groups, a vast generalisation of the notion of braid groups, an important and ubiquitous class of groups. Despite their introduction in the 1970s in connection with problems from algebraic geometry, Artin groups are still mysterious in many ways. A central problem at the heart of this project is the so-called “Isomorphism Problem”: Artin groups are defined by means of graphs, and the Isomorphism Problem asks to determine exactly when two different graphs yield the same Artin group. This important problem is currently wide open for Artin groups, and it remains open even for the related -and better understood- class of Coxeter groups. Solving this problem, and more generally finding isomorphism invariants for Artin groups, would have far-reaching consequences for our understanding of these groups. The goal of this project is to develop a geometric framework to solve the Isomorphism Problem for new classes of Artin groups. It will vastly expand on the Project Lead’s previous work on the geometry of Artin groups, and will involve constructing new spaces on which Artin groups act. In particular, we propose a programme to solve a recent conjecture on the hyperbolicity of certain “curve graphs” associated to large classes of Artin groups. A solution to this conjecture would have several striking consequences for the structure of these groups, and we will in particular use the geometry of these graphs to solve the Isomorphism Problem for large classes of Artin groups. The geometric framework at the heart of this project is general enough to be applied beyond the world of Artin groups. In particular, we plan to study many related classes of groups from that perspective, including Coxeter groups, graph products of groups, and groups associated with hyperplane arrangements. The unifying approach pioneered by this project will introduce exciting new geometric perspectives to tackle long-standing open problems about these groups, and will strengthen the existing connections between these important classes of groups.