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

Jingyin Huang

TL;DR

The paper develops a geometrically robust framework to analyze cycles in spherical Deligne complexes Δ_S of spherical Artin groups by isolating large non-positively curved subcomplexes (islands) and using relative Artin complexes Δ_{S,S'}. It integrates subdivision and bowtie/flag criteria to derive contractibility results for cores, enabling transfer of K(π,1) from spherical/parabolic subgroups to the ambient Artin group. This yields new K(π,1) results: (i) for all 3-dimensional hyperbolic Artin groups except one, (ii) for all quasi-Lannér hyperbolic types up to dimension 4, and (iii) for Artin groups with complete bipartite Coxeter diagrams in higher dimensions; it also provides a cascade of structural tools—Falk complexes, injective metrics, and folded Artin complexes—to analyze hyperplane arrangements and their relation to Artin complexes. Collectively, these methods advance the K(π,1) program for broad families of Artin groups and illuminate how local non-positively curved pieces control global asphericity and group-theoretic properties. The results have potential implications for word problems, parabolic subgroups, and stability phenomena in Artin groups, tying Deligne complex geometry to algebraic structure through explicit curvature-aware decompositions.

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 $K(π,1)$-conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lannér hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the $K(π,1)$-conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond.

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

TL;DR

The paper develops a geometrically robust framework to analyze cycles in spherical Deligne complexes Δ_S of spherical Artin groups by isolating large non-positively curved subcomplexes (islands) and using relative Artin complexes Δ_{S,S'}. It integrates subdivision and bowtie/flag criteria to derive contractibility results for cores, enabling transfer of K(π,1) from spherical/parabolic subgroups to the ambient Artin group. This yields new K(π,1) results: (i) for all 3-dimensional hyperbolic Artin groups except one, (ii) for all quasi-Lannér hyperbolic types up to dimension 4, and (iii) for Artin groups with complete bipartite Coxeter diagrams in higher dimensions; it also provides a cascade of structural tools—Falk complexes, injective metrics, and folded Artin complexes—to analyze hyperplane arrangements and their relation to Artin complexes. Collectively, these methods advance the K(π,1) program for broad families of Artin groups and illuminate how local non-positively curved pieces control global asphericity and group-theoretic properties. The results have potential implications for word problems, parabolic subgroups, and stability phenomena in Artin groups, tying Deligne complex geometry to algebraic structure through explicit curvature-aware decompositions.

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 -conjecture holds for all 3-dimensional hyperbolic type Artin groups, except one single example; and the conjecture holds for all quasi-Lannér hyperbolic type Artin groups up to dimension 4. In higher dimension, we show the -conjecture for Artin groups whose Coxeter diagrams are complete bipartite (edge labels can be arbitrary), answering a question of J. McCammond.
Paper Structure (39 sections, 86 theorems, 65 equations, 17 figures)

This paper contains 39 sections, 86 theorems, 65 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Cycles in some spherical Deligne complexes
  4. Application to the $K(\pi,1)$-conjecture
  5. Non-positive curvature pieces in spherical Deligne complexes
  6. Structure of the article and reading guide
  7. Acknowledgment
  8. $K(\pi,1)$-conjecture via relative Artin complexes
  9. Artin complexes and relative Artin complexes
  10. $K(\pi,1)$-conjecture via relative Artin complexes
  11. Contractibility of core
  12. Two contractibility criterion for simplicial complexes
  13. Subdivision of some relative Artin complexes
  14. Link conditions at the base case
  15. One side flagness in type $F_4$
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose for any irreducible spherical Artin groups $A_S$, any loop of length $<2\pi$ in the associated spherical Deligne complex $\Delta_S$ with Moussong metric has a length non-increasing homotopy to the trivial loop. Then the $K(\pi,1)$-conjecture holds for any Artin group.

Figures (17)

  • Figure 1: Spherical and affine diagrams: unlabeled edges are assumed to have label 3.
  • Figure 2: Some examples: unlabeled edges are assumed to have label 3.
  • Figure 3: Diagrams of type $\widetilde{B}$ and $\widetilde{D}$.
  • Figure 4: Coxeter diagram of type $D_n$.
  • Figure 5: Auxiliary sub-arrangement I.
  • ...and 12 more figures

Theorems & Definitions (163)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 153 more