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Classification of Artin groups admitting retractions onto their parabolic subgroups

Bruno Aarón Cisneros de la Cruz, María Cumplido, Islam Foniqi, Luis Paris

Abstract

We classify the Artin groups that admit retractions onto all of their parabolic subgroups. Our approach relies on a detailed analysis of triangular subgroups, with a key ingredient being the classification of homomorphisms between dihedral Artin groups that map one of the standard generators to a standard generator. As a consequence, we show that whenever an Artin group admits retractions to parabolic subgroups, it also admits ordinary ones - that is, retractions that send each standard generator either to a standard generator or to the identity.

Classification of Artin groups admitting retractions onto their parabolic subgroups

Abstract

We classify the Artin groups that admit retractions onto all of their parabolic subgroups. Our approach relies on a detailed analysis of triangular subgroups, with a key ingredient being the classification of homomorphisms between dihedral Artin groups that map one of the standard generators to a standard generator. As a consequence, we show that whenever an Artin group admits retractions to parabolic subgroups, it also admits ordinary ones - that is, retractions that send each standard generator either to a standard generator or to the identity.
Paper Structure (7 sections, 17 theorems, 79 equations, 4 figures)

This paper contains 7 sections, 17 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Classification of homomorphisms between dihedral Artin groups
  4. Bass-Serre Tree
  5. A Lemma
  6. Case Analysis - Proof of \ref{['theorem:classification_of_homo']}
  7. Characterizing Parabolic-Retractable Artin Groups

Key Result

Theorem 1.1

Let $A_S$ be any Artin group. Then $A_S$ is parabolic-retractable if and only if, for any triple $a, b, c \in S$ of pairwise distinct elements, $A_{a,b,c}$ retracts onto $A_{b,c}$.

Figures (4)

  • Figure 1: An Artin group with a retract-compatible Coxeter matrix.
  • Figure 2: A parabolic-retractable Artin group.
  • Figure 3: Another Artin group with a retract-compatible Coxeter matrix whose labels are not ordered by division.
  • Figure :

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 21 more