A Deligne complex for Artin Monoids

Abstract

In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis [CD95a] to study the K(π, 1) conjecture for these groups. Using a notion of Artin monoid cosets introduced by the first author in [Boyd20], we construct a version of the Deligne complex for Artin monoids.We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.

La conjecture du $K(π,1)$ pour les groupes d'Artin affines (d'après Paolini et Salvetti)

“…Charney et Davis ont muni le complexe de Deligne d'une métrique CAT(0) dans certains cas, ce qui leur permet de montrer sa contractibilité (Hendriks, 1985 ;Charney et Davis, 1995b). Cette stratégie de munir le complexe de Deligne d'une métrique à courbure négative est l'objet de plusieurs travaux récents (Charney, 2004 ;Godelle, 2007 ;Boyd, Charney et Morris-Wright, 2020 ;Haettel, 2021 ;Goldman, 2021 ;Morris-Wright, 2021).…”