Workshop

Braid groups are the original Artin groups and they have many different classifyng spaces: (1) Quotient of a complex hyperplane complement, (2) Monic polynomials with distinct roots, (3) Salvetti complex from face-identifying a permutahedron, and (4) Dual braid complex from face-identifying the order complex of the noncrossing partition lattice. This talk connects the hyperplane complement and the monic polynomials to the noncrossing partitions in the dual structure.  It is based on joint work with Michael Dougherty.

The theory of arrangements recently broadened its scope beyond the linear case to include arrangements in the torus, in products of elliptic curves and, more generally, in Abelian Lie groups. The classical K(pi,1) problem, i.e. deciding asphericity of an arrangement's complement in combinatorial terms, can be stated in general. In this talk I will review some of the classical history of this problem and present some recent advances in the non-linear case. The talk will report on joint work with Christin Bibby, Alessio D'Alì, Noriane Girard, Giovanni Paolini, Sonja Riedel.

In this talk, I will discuss profinite properties of Artin groups, such as residual finiteness, and the connection between the group and its profinite completion. Informally, those properties tell us if the geometry and the algebra of the group can be approximated by its finite quotients. I will discuss some of my contributions, including joint work with Kevin Schreve. 

In the last 30 years, geometric representation theory has produced a large stock of examples of linear 2-representations of Artin groups.  These 2-representations have not really been used yet to prove genuinely new theorems about the groups themselves, but there is good reason to believe that they will eventually be useful to geometric group theorists.  In the first part of this talk I will describe a simple example of a 2-representation of an Artin group, and explain how much of the classical theory of Artin groups is visible through it.  In the latter part of the talk I will try to explain an approach to the K(\pi,1) conjecture which emerges from the study of spherical objects in this simple 2-representation.

A chambered set for a Coxeter system is a set with an action by the Coxeter group and a specified set of orbit representatives, called the fundamental chamber, the stabilizer of each point of which is a standard parabolic subgroup. This notion arises naturally in describing dependence of facets of subarrangements of the Coxeter arrangement on the ambient space of the arrangement, which could be the Tits cone, Coxeter complex, Davis complex, imaginary cone, Cayley graph etc. This talk will discuss a monoidal category of chambered sets, and related structure. For finite Coxeter groups, part of this structure categorifies the descent algebra and certain modules for it, as studied by Solomon, Tits and Moszkowski. The results of Solomon and Tits for general Coxeter groups imply that the descent algebra has no satisfactory analogue, as an algebra, in general; the monoidal category of chambered sets provides a refined alternative.

Recall that an Artin group is said of spherical type if its associated Coxeter group is finite. This talk is a general presentation on the automorphism groups of Artin groups of spherical type. We will present known and somewhat lesser-known (because unpublished) results on these automorphism groups, as well as open questions concerning them.

While the isomorphism problem is undecidable in the full universe of groups, the region containing Coxeter groups is fairly well understood and it is expected that the problem is solvable within this class of groups. The Coxeter galaxy provides a framework to study this and related, more refined questions. In this talk we will provide an overview on the current state of the art concerning this problem and present some new results obtained in joint work with Yuri Santos Rego. 

I shall present a general definition of what should be considered a parabolic subgroup for the generalized braid group associated to a complex reflection group, and present a series of remarkable properties of these subgroups. These properties generalize the ones obtained earlier by Cumplido, Gebhardt, González-Meneses and Wiest for real groups, with a combinatorial definition of parabolic subgroups attached to a specific (Artin) presentation. First of all, intersection of parabolic subgroups are parabolic subgroups; this implies that they form a lattice of subgroups. Secondly, they are also the vertices of a graph on which the generalized braid group acts faithfully (modulo center), and which generalizes the curve graph for the usual braid group on n strands. As a consequence, this graph is conjectured to be hyperbolic. This is joint work with J. González-Meneses (Sevilla, Spain) for the most part -- only one reflection group remained untractable using our methods, a difficulty which already appeared in the proof of the K(\pi,1) conjecture for these groups. This last case has been settled recently by my student O. Garnier.

It has long been recognized that there are many fruitful connections between the protagonists of this workshop: hyperplane arrangements and Artin groups. I will discuss in this talk a connection between two kinds of fibrations that occur in these contexts. One is the Milnor fibration of the complement of an arrangement of complex hyperplanes (perhaps taken with multiplicities) over the circle. The other is an "algebraic fibration" of the right-angled Artin group associated to a graph (perhaps with weights assigned to the vertices) over the integers. In either case, one is interested in the topological invariants of the fiber (or the kernel), be it the Milnor fiber of the (multi-)arrangement, the Bestvina-Brady group of the graph, or the Artin kernel of the weighted graph. After a brief overview of these topics, I will focus on the case when the monodromy of the aforementioned fibrations is trivial in homology, and I will outline several new results in this direction, some of which can be stated in a unified way.