On spherical Deligne complexes of type $D_n$

TL;DR

This paper investigates the structure of 6-cycles in the spherical Deligne complex of type D_n, identifying centers or quasi-centers, which aids in proving the K(π,1)-conjecture for certain Artin groups and shows some complexes are CAT(1).

Contribution

It introduces a detailed analysis of 6-cycle configurations in the spherical Deligne complex of type D_n, crucial for advancing the K(π,1)-conjecture proof.

Findings

Certain 6-cycles have a center or quasi-center

A 2-dimensional relative Artin complex is CAT(1)

Results support K(π,1)-conjecture for specific Artin groups

Abstract

Let $\Delta$ be the Artin complex of the Artin group of type $D_n$. This complex is also called the spherical Deligne complex of type $D_n$. We show certain types of 6-cycles in the 1-skeleton of $\Delta$ either have a center, which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center, which is a vertex adjacent to three of the alternating vertices of the 6-cycle. This will be a key ingredient in proving $K(\pi,1)$-conjecture for several classes of Artin groups in a companion article. As a consequence, we also deduce that certain 2-dimensional relative Artin complex inside the $D_n$-type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$.