A Deligne complex for Artin Monoids

TL;DR

This paper introduces a geometric cube complex called the Deligne complex for Artin monoids, proving its contractibility and exploring its embedding properties into the Artin group Deligne complex, with implications for CAT(0) geometry.

Contribution

It constructs a Deligne complex for Artin monoids, proves its contractibility, and analyzes its embedding into the Artin group Deligne complex, including CAT(0) properties.

Findings

The monoid Deligne complex is contractible.

Embedding into the Artin group Deligne complex is locally isometric.

For FC-type Artin groups, the embedding is globally isometric and the complex is CAT(0).

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 (1995) to study the K(\pi,1) conjecture for these groups. Using a notion of Artin monoid cosets, 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.