A Deligne complex for Artin monoids

Boyd, R. , Charney, R. and Morris-Wright, R. (2022) A Deligne complex for Artin monoids. Journal of Algebra, 607(B), pp. 53-78. (doi: 10.1016/j.jalgebra.2021.03.015)

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 [8] to study the 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.

• Item Type: Articles
• Additional Information: Charney was partially supported by NSF grant DMS-1607616. Boyd was partially supported by the London Mathematical Society Cecil King Scholarship.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Boyd, Dr Rachael
• Authors: Boyd, R., Charney, R., and Morris-Wright, R.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Algebra
• Publisher: Elsevier
• ISSN: 0021-8693
• ISSN (Online): 1090-266X
• Published Online: 31 March 2021
• Copyright Holders: Copyright © 2022 Elsevier
• First Published: First published in Journal of Algebra 607(B):53-78
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher