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Bestvina metric and tree reduction for $K(π,1)$-conjecture

Jingyin Huang

Abstract

We reduce the $K(π,1)$-conjecture for all Artin groups to properties of Artin groups whose Coxeter diagrams are trees, from which we deduce new classes of Artin groups satisfying the $K(π,1)$-conjecture. This relies on constructing actions of Artin groups on Bestvina complexes of suitable Garside groupoids.

Bestvina metric and tree reduction for $K(π,1)$-conjecture

Abstract

We reduce the -conjecture for all Artin groups to properties of Artin groups whose Coxeter diagrams are trees, from which we deduce new classes of Artin groups satisfying the -conjecture. This relies on constructing actions of Artin groups on Bestvina complexes of suitable Garside groupoids.
Paper Structure (37 sections, 89 theorems, 28 equations, 5 figures)

This paper contains 37 sections, 89 theorems, 28 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Discussion of proofs
  4. Structure of the article
  5. Acknowledgment
  6. Complexes for Artin and Coxeter groups
  7. Artin complexes and relative Artin complexes
  8. Davis complexes
  9. Oriented Davis complexes and Salvetti complexes
  10. Generalized cycles
  11. Relation on sets
  12. Posets
  13. Partial cyclic order
  14. Minimal cuts in graphs
  15. The labeled 4-cycle condition
  16. ...and 22 more sections

Key Result

Theorem 1.1

(Corollary cor:cycle reduction all) Suppose that for each Artin group $A_S$ with Coxeter diagram being a tree, $A_S$ satisfies the $K(\pi,1)$-conjecture and any special 4-cycle in $\Delta_S$ has a center. Then any Artin group satisfies the $K(\pi,1)$-conjecture.

Figures (5)

  • Figure 1: The diagram $\Lambda$.
  • Figure 2: Two vertices are joined by an edge if they satisfy the relation $\sim$.
  • Figure 3: A strip.
  • Figure 4: The disk $D$.
  • Figure 5:

Theorems & Definitions (177)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 167 more