Note on the residual finiteness of Artin groups

Abstract

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 admissible partition {\mathcal{P}} determines a quotient Coxeter graph {\Gamma/\mathcal{P}} . We prove that, if {\Gamma/\mathcal{P}} is either a forest or an even triangle free Coxeter graph and {A_{X}} is residually finite for all {X\in\mathcal{P}} , then A is residually finite.

On the isomorphism problem for even Artin groups

“…The two most studied and best understood families of Artin groups are the one of Artin groups of spherical type and the one of right-angled Artin groups. The family of even Artin groups has not been so much studied (see, however, Blasco-García-Cogolludo-Agustín [3], Blasco-García-Juhász-Paris [4] and Blasco-García-Martínez-Pérez-Paris [5]), but we are convinced by its interest, in particular because even Artin groups share several interesting properties with right-angled Artin groups. For instance, any even Artin group retracts into any of its parabolic subgroups.…”

Poly-freeness in large even Artin groups

“…An Artin group is called even if m e is an even number for all e ∈ E. There are not many results in the literature about even Artin groups (but see [3], [2] and [1]). However, we think that this family deserves more attention since they have remarkable properties some of them shared with the familiy of right-angled Artin groups.…”

Residual finiteness of certain 2-dimensional Artin groups

“…• for each distinct X, Y ∈ P there is at most one edge joining a a vertex of X with a vertex of Y in Γ, • the graphs Γ/P is either a forest, or a triangle free graph with even labels, are also residually finite by (Blasco-Garcia)-Juhasz-Paris [BGJP18]. The residual finiteness of 3generator affine Artin groups, i.e.…”