Cycles in spherical Deligne complexes and application to $K(\pi,1)$-conjecture for Artin groups

Abstract

We introduce a method of finding large non-positively curved subcomplexes in certain spherical Deligne complexes, which is effective for studying fillings of certain 6-cycles in spherical Deligne complexes. As applications, we show the $K(\pi,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lann\'er hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the $K(\pi,1)$-conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond.