Related papers: On a theorem of Artin

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_\Gamma$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_\Gamma)$ virtually surjects onto some infinite Coxeter group. In particular,…

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y \in \mathcal{P}$, $X \neq Y$, there is at most one pair $(s,t) \in X \times Y$ which has a relation. An…

We study fixed subgroups of automorphisms of any large-type Artin group $A_{\Gamma}$. We define a natural subgroup $\mathrm{Aut}_\Gamma(A_\Gamma)$ of $\mathrm{Aut}(A_{\Gamma})$, and for every $\gamma \in \mathrm{Aut}_\Gamma(A_\Gamma)$ we…

A finite simplicial graph \Gamma determines a right-angled Artin group G_\Gamma, with generators corresponding to the vertices of \Gamma, and with a relation vw=wv for each pair of adjacent vertices. We compute the lower central series…

Consider an affine Coxeter group $W$ acting by isometries on the Euclidean space $\mathbb{R}^n$, and the arrangement of its reflection hyperplanes. The fundamental group of the complement $Y_W$ of the complexification of this arrangement in…

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…

We study two global structural properties of a graph $\Gamma$, denoted AS and CFS, which arise in a natural way from geometric group theory. We study these properties in the Erd\"os--R\'enyi random graph model G(n,p), proving a sharp…

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we…

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…

Let $\Gamma$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,\Sigma$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and…

Given a graph $\Gamma$, the right-angled Artin group $A(\Gamma)$ is given by the presentation $\langle u \in V(\Gamma) \mid [u,v]=1, \ \{u,v\} \in E(\Gamma) \rangle$. The Embedding Problem in right-angled Artin groups asks, given two finite…

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…

Let $N$ be a closed nonorientable surface with or without marked points. In this paper we prove that, for every finite full subgraph $\Gamma$ of $\mathcal{C}^{\mathrm{two}}(N)$, the right-angled Artin group on $\Gamma$ can be embedded in…

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for…

For all Artin groups, we characterise the girth (i.e. the length of a shortest cycle) of the defining graph algebraically, showing that it is an isomorphism invariant. Using this result, we prove that the Artin groups based on a cycle graph…

Let $\Gamma$ be a simplicial, finite, connected graph such that $\Gamma$ does not decompose as a nontrivial join. We prove that two notions of strong quasiconvexity and stability are equivalent in the right-angled Artin group $A_\Gamma$…

Let $A_\Gamma$ be an Artin group with defining graph $\Gamma$. We introduce the notion of $A_\Gamma$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_\Gamma$ being extra-large…

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(\Gamma)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In…

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is…

Virtual Artin groups were recently introduced by Bellingeri, Paris, and Thiel as broad generalizations of the well-known virtual braid groups. For each Coxeter graph $\Gamma$, they defined the virtual Artin group $VA[\Gamma]$, which is…