Virtually Cocompactly Cubulated Artin–Tits Groups

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2. We extend a well‐known result that Coxeter groups are to a class of Shephard groups that have ‘enough’ finite parabolic subgroups. We also show that in this setting, if the associated Coxeter group is type (FC), then the Shephard group acts properly and cocompactly on a cube complex. As part of our proof of the former result, we introduce a new criteria for a complex made of simplices to be .

We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum–Connes conjecture with coefficients.

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.

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{{CAT}}\\,}}(0)$$\\end{document} cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.

On loxodromic actions of Artin–Tits groups

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

Configuration Lie groupoids and orbifold braid groups

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

For any right-angled Artin group $A_Γ$ we construct a finite-dimensional space $\mathcal{O}_Γ$ on which the group $\text{Out}(A_Γ)$ of outer automorphisms of $A_Γ$ acts with finite point stabilizers. We prove that $\mathcal{O}_Γ$ is contractible, so that the quotient is a rational classifying space for $\text{Out}(A_Γ)$. The space $\mathcal{O}_Γ$ blends features of the symmetric space of lattices in $\mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $\mathcal{O}_Γ$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with $A_Γ$.

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

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.

Motivated by the computation of local system cohomology of Artin groups (and their configuration spaces) we introduce some very special class of sheaves over posets, called weighted sheaves over posets, which seem to be interesting by themselves. We relate them with some graph complexes arising from Artin groups. In particular, we find an interesting connection between some local cohomology of the braid groups, localized at the d-cyclotomic polynomial, and the cohomology of what we call d-independent graph complexes.

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

BMW algebras of simply laced type

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples that come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov that asserts that the rank gradient of infinite boundedly generated residually finite groups is zero. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.