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Centers of Artin groups defined on cones

Kasia Jankiewicz, MurphyKate Montee

TL;DR

The paper studies the Center Conjecture for Artin groups defined by graphs with cone points and proves that the conjecture passes from the cone-point subgroups $A_T$ to the entire group $A_\Gamma$, yielding new cases such as graphs with a single cone point. It establishes $Z(A_\Gamma)\subseteq Z(A_T)$ via amalgamated-product decompositions, so trivial centers of $A_T$ force trivial centers of $A_\Gamma$, recovering known results when $T=\emptyset$. A retraction map $\pi_X$ is developed to constrain central elements, showing that nontrivial central elements must interact with cone-point subsets in a way compatible with spherical factors. Consequently, if $A_T$ satisfies the Center Conjecture, then $A_\Gamma$ does as well, providing new classes of graphs for which the conjecture holds and generalizing previous results to broader cone-set configurations.

Abstract

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.

Centers of Artin groups defined on cones

TL;DR

The paper studies the Center Conjecture for Artin groups defined by graphs with cone points and proves that the conjecture passes from the cone-point subgroups to the entire group , yielding new cases such as graphs with a single cone point. It establishes via amalgamated-product decompositions, so trivial centers of force trivial centers of , recovering known results when . A retraction map is developed to constrain central elements, showing that nontrivial central elements must interact with cone-point subsets in a way compatible with spherical factors. Consequently, if satisfies the Center Conjecture, then does as well, providing new classes of graphs for which the conjecture holds and generalizing previous results to broader cone-set configurations.

Abstract

We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.
Paper Structure (5 sections, 10 theorems, 7 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Central Elements are Generated by Cone Points
  3. Cone Sets Satisfying the Center Conjecture
  4. Retraction map
  5. Proof of \ref{['thm intro: center conjectures passes to cones']}

Key Result

Proposition 0

Let $A_\Gamma$ be an Artin group, where $T\subseteq V(\Gamma)$ is the set of cone points of $\Gamma$. Then $Z(A_{\Gamma})\subseteq Z(A_T)$.

Figures (1)

  • Figure 1: New examples of graphs whose associated Artin groups satisfy the Center Conjecture. The red subgraphs are spanned by the set of cone points. The middle graph can have any labels. The diagram on the right represents an infinite class of such graphs; the double edges represent joins, with all labels equal to 3.

Theorems & Definitions (15)

  • Proposition 0
  • Theorem 1
  • Lemma 2
  • Proposition 2
  • proof
  • Corollary 3
  • Corollary 4
  • proof
  • Lemma 5: Lemma 2.3 GodelleParis12
  • Definition 6: BlufsteinParis23
  • ...and 5 more