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From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics

Nima Hoda, Jingyin Huang

Abstract

We reduce the $K(π,1)$-conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the $K(π,1)$-conjecture holds for every Artin group whose spherical parabolic subgroups avoid type $D_n$ ($n \ge 4$) and the exceptional types. The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro.

From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics

Abstract

We reduce the -conjecture for all Artin groups with tree Coxeter diagrams to properties of Artin groups with tripod-shaped Coxeter diagrams. Combining this reduction theorem and properties of braid groups in previous works of Charney, Crisp-McCammond, Haettel and the second named author, we deduce that the -conjecture holds for every Artin group whose spherical parabolic subgroups avoid type () and the exceptional types. The reduction theorem relies on producing a ``tower'' of injective metric spaces from a single Artin group. The construction of such a tower relies on two ingredients of independent interests: a notion of combinatorial convexity and a Bestvina-type inequality, in certain injective orthoscheme complexes. These ingredients further rely on the use of structural properties of bi-Helly graphs (also known as absolute bipartite retracts) developed in joint work of the first named author with Munro.
Paper Structure (28 sections, 72 theorems, 24 equations, 9 figures)

This paper contains 28 sections, 72 theorems, 24 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A reduction theorem for the $K(\pi,1)$-conjecture
  3. Applications
  4. Discussion of proof
  5. Organization of the article
  6. Acknowledgment
  7. Preliminaries
  8. Artin complexes and relative Artin complexes
  9. Posets
  10. Poset properties of simplicial complexes of type $S$
  11. A contractibility criterion for simplicial complexes
  12. Normal forms and combinatorial convexity on orthoscheme complexes
  13. Bi-Helly graphs and normal forms
  14. Bi-Helly subgraphs of $\widetilde{C}$-like complexes
  15. Normal forms on orthoscheme complexes
  16. ...and 13 more sections

Key Result

Theorem 1.2

If Conjecture conj:braid holds, then the $K(\pi,1)$-conjecture holds for any Artin group whose every irreducible spherical parabolic subgroup is of type $A$, $B$, or $I_2$.

Figures (9)

  • Figure 1: Three special families.
  • Figure 2: $\widetilde{B}$-like and $\widetilde{D}$-like subdiagrams.
  • Figure 3: The thickened subdiagram indicates the robust $\widetilde{C}$-core $\Lambda'$. In (3), $\Lambda'$ can be $\widetilde{D}_4$-like, in which case $b_m$ is a leaf vertex of $\Lambda'$. Similarly, in (4), $\Lambda'$ can be $\widetilde{B}_3$-like, in which case $b_m$ is a leaf vertex of $\Lambda'$.
  • Figure 4: Some diagrams in the proof of Lemma \ref{['lem:corner1']}.
  • Figure 5: Some diagrams in the proof of Proposition \ref{['prop:intersection']}.
  • ...and 4 more figures

Theorems & Definitions (139)

  • Conjecture 1.1: Charney--Davis
  • Theorem 1.2: CharneyDavischarney2004deligne
  • Theorem 1.3
  • Theorem 1.4: huangbestvina
  • Definition 1.5
  • Theorem 1.6: Informal version of the main theorem
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 129 more