Configuration Lie groupoids and orbifold braid groups

A four-term exact sequence of surface orbifold pure braid groups

Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(\pi,1) -conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland’s stability conditions provide new realisations of the corresponding K(\pi,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(\pi,1) -conjecture.

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.

We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

Jacobi structures and Spencer operators

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on \mathbb C^n is known to be a {\mathcal K}(\pi ,1) space for the corresponding Artin group \mathcal A . A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type \tilde{\mathcal A}_n and its associated Artin group, the affine braid group \tilde{\mathcal A} . Using the fact that \tilde{\mathcal A} can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, n -dimensional {\mathcal K}(\pi ,1) -space for \tilde{\mathcal A} , and use it to prove the {\mathcal K}(\pi ,1) -conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.

BMW algebras of simply laced type

The singular braids with [Formula: see text] strands, [Formula: see text], were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by [Formula: see text]. There has been another generalization of braid groups, denoted by [Formula: see text], [Formula: see text], which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group [Formula: see text] simultaneously generalizes the classical braid group, as well as the virtual braid group on [Formula: see text] strands. We investigate the commutator subgroups [Formula: see text] and [Formula: see text] of these generalized braid groups. We prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated if and only if [Formula: see text]. Further, we show that both [Formula: see text] and [Formula: see text] are perfect if and only if [Formula: see text].

Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system A_n ; in the second part we prove a similar theorem for the root system D_n .

Braid groups were introduced by Artin [1]. These groups have been studied extensively—see [2], [9] and the references cited there. Recently work has been done on “circular” braid groups and other “braid-like” groups [7], [10]. In this paper we formulate the concept of a generalized braid group, and we begin a study of the structure of such groups. In particular for such a group G1, there is a homomorphism from G onto the infinite cyclic group, the kernel of which is the derived group G1 of G. We study G1. Our results generalize results of Gorin and Lin [5], who considered the case when G is a classical braid group B(n ≥ 3). They showed that is free abelian of rank 2 if n = 3, 4 and is trivial if n ≥ 5. They also showed that is finitely presented.

Understanding the lower central series of a group is, in general, a difficult task. It is, however, a rewarding one: computing the lower central series and the associated Lie algebras of a group or of some of its subgroups can lead to a deep understanding of the underlying structure of that group. Our goal here is to showcase several techniques aimed at carrying out part of this task. In particular, we seek to answer the following question: when does the lower central series stop? We introduce a number of tools that we then apply to various groups related to braid groups: the braid groups themselves, surface braid groups, groups of virtual and welded braids, and partitioned versions of all of these groups. The path from our general techniques to their application is far from being a straight one, and some astuteness and tenacity is required to deal with all of the cases encountered along the way. Nevertheless, we arrive at an answer to our question for each and every one of these groups, save for one family of partitioned braid groups on the projective plane. In several cases, we even compute completely the lower central series. Some results about the lower central series of Artin groups are also included.

The distribution of hapten binding enthalpies in conventionally-raised antibody populations

Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections.The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra.For each irreducible representation of G the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the G-orbits of regular points in the complexification of the Euclidean space.The fundamental group is the generalized Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups.