Related papers: $K(\pi,1)$ and word problems for infinite type Art…

In this doctoral thesis, we will determine the image of Artin groups associated to all finite irreducible Coxeter groups inside their associated finite Iwahori-Hecke algebra. This was done in type $A$ by Brunat, Magaard and Marin. The…

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four…

We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this…

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special…

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are…

Let $\Sigma$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $\Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The…

Dual presentations of Coxeter groups have recently led to breakthroughs in our understanding of affine Artin groups. In particular, they led to the proof of the $K(\pi, 1)$ conjecture and to the solution of the word problem. Will the "dual…

The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type…

Let $(W,S)$ be a Coxeter system, $S$ finite, and let $G_{W}$ be the associated Artin group. One has configuration spaces $Y,\ Y_{W},$ where $G_{W}=\pi_1(Y_{W}),$ and a natural $W$-covering $f_{W}:\ Y\to Y_{W}.$ The Schwarz genus $g(f_{W})$…

Let $(W,S)$ be a Coxeter system and $\Gamma$ be a group of automorphisms of $W$ such that $\gamma(S)=S$ for all $\gamma \in \Gamma$. Then it is known that the group of fixed points $W^\Gamma$ is again a Coxeter group with a canonically…

We introduce the universal virtual braid group $UV_n(c)$, which provides a unified algebraic framework for virtual braid--type structures with $c$ types of crossings and admits natural quotient maps onto the standard families in the…

We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with…

We extend the theory of dual Coxeter and Artin groups to all rank-three Coxeter systems, beyond the previously studied spherical and affine cases. Using geometric, combinatorial, and topological techniques, we show that rank-three…

We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy…

Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$…

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander-Markov correspondence for the theory of stable isotopy classes of…

We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled…

Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\reg}$ is a $K(\pi,1)$ space. This…

We prove universal (case-free) formulas for the weighted enumeration of factorizations of Coxeter elements into products of reflections valid in any well-generated reflection group $W$, in terms of the spectrum of an associated operator,…

There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (\Gamma) and a subgraph (\Lambda) of its complement, checking some conditions, and, if satisfied, producing a graph (\Delta) such that the right-angled…