My project aims at proving the usefulness of categorical methods for questions in geometric group theory. Groups are central objects in mathematics: integers form a group under addition, symmetries of an object form a group,... The idea of looking at a group as a geometric object goes back to Gromov in the late 80's, and has had remarkable consequences. If looked at from far away, groups turn out to have behavior that fall in different classes, which in turn govern some global features of the groups. Amongst groups, braid groups have received a large amount of attention, notably because of their appearance in several research contexts (topology, algebra, quantum physics,...). However, despite all this attention, we know surprisingly little about them (and their generalization), in particular from the geometric point of view.
Categories on the other hand are algebraic structures that were introduced in the middle of the twentieth century. They somehow reflect the idea that often, what one cares about is not so much the objects themselves, but rather the ways to relate them -- what's called morphisms between them. These ideas have helped shape and organize a good part of modern algebra. Lately, these concepts met ideas from quantum physics, where researchers were trying to upgrade quantum field theories in dimension 3 (2 in space, 1 in time) to dimension 4 (3+1). A mathematical reformulation suggested to look for groups acting on categories rather than on classical spaces. In particular, Khovanov and Seidel introduced in 2000 an action of the braid groups on a certain category that is quite simple and has interesting features.
The goal of my project is to use this latter Khovanov-Seidel tool to better understand the geometry of braid groups and their extensions (called Artin-Tits groups). Classical approaches often rely on the identification of braids with the symmetries of a space of curves. It appears that Khovanov-Seidel's model is consistent with this approach whenever this makes sense, but can be defined in a much broader generality, which is very promising.
Two questions have been identified as main goals: studying the faithfulness of the so-called Burau representation, and studying the Haagerup property. The first one has been open for almost a century and is one of the most intriguing questions about braids. Proving it to be false would have surprising consequences in the theory of knots, and it being right would be a key step towards other algebraic conjectures. It should also be mentioned that the Burau representation is some kind of shadow of Khovanov-Seidel's construction, which justifies our approach.
The Haagerup property is something very geometric, describing the way braids can, or cannot, act on some large spaces (Hilbert spaces). It is surprising how little we know about this question: an answer is only known in the case of the 3-strand braid group. Finding a way to approach this question from category theory would be quite striking, provide new tools for geometers, but also suggest relevant geometries in the context of category theory that would be of great interest.
