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

TL;DR

This paper develops a method to find non-positively curved subcomplexes in spherical Deligne complexes, leading to significant progress on the $K(,)$-conjecture for various classes of Artin groups, especially in low dimensions.

Contribution

It introduces a new approach to identify non-positively curved subcomplexes in spherical Deligne complexes, advancing the understanding of the $K(,)$-conjecture for hyperbolic Artin groups.

Findings

Proves the $K(,)$-conjecture for all 3-dimensional hyperbolic type Artin groups except one.

Establishes the conjecture for all quasi-Lanner hyperbolic type Artin groups up to dimension 4.

Confirms the conjecture for Artin groups with complete bipartite Coxeter diagrams in higher dimensions.

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.