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Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups (2405.12068v3)

Published 20 May 2024 in math.GR, math.GT, and math.MG

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.

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  1. Jingyin Huang
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