On spherical Deligne complexes of type $D_n$
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 6cycles in the 1skeleton of $\Delta$ either have a center, which is a vertex adjacent to each vertex of the 6cycle, or a quasicenter, which is a vertex adjacent to three of the alternating vertices of the 6cycle. 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 2dimensional relative Artin complex inside the $D_n$type Artin complex, endowed with the induced Moussong metric, is CAT$(1)$.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.11374
 arXiv:
 arXiv:2405.11374
 Bibcode:
 2024arXiv240511374H
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Geometric Topology
 EPrint:
 50 pages, 12 figures. Add a reference