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Subgroup Separability of Artin Groups II

Kisnney Almeida, Igor Lima

Abstract

In this paper, we establish a new criterion for determining whether an Artin group is subgroup separable (LERF), building upon a criterion introduced in a previous work. Specifically, we prove an Artin group is LERF if and only if it does not contain certain induced subgraphs, thus providing a more direct generalization of the Metaftsis-Raptis criterion for RAAGs. As consequences, we prove an Artin group is ERF if and only if it is a free abelian group and we establish a link between subgroup separability and coherence for Artin groups.

Subgroup Separability of Artin Groups II

Abstract

In this paper, we establish a new criterion for determining whether an Artin group is subgroup separable (LERF), building upon a criterion introduced in a previous work. Specifically, we prove an Artin group is LERF if and only if it does not contain certain induced subgraphs, thus providing a more direct generalization of the Metaftsis-Raptis criterion for RAAGs. As consequences, we prove an Artin group is ERF if and only if it is a free abelian group and we establish a link between subgroup separability and coherence for Artin groups.
Paper Structure (4 sections, 10 theorems, 2 equations)

This paper contains 4 sections, 10 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['teopois']}
  4. Applications

Key Result

Theorem 1.1

Let $A=A(\Gamma)$ be an Artin group. Then $A$ is subgroup separable if and only if $\Gamma$ is not poisonous.

Theorems & Definitions (14)

  • Definition
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: S,S2
  • Theorem 2.2: AG, Th. 4
  • Theorem 2.3
  • Lemma 2.4
  • proof : Proof of Theorem \ref{['teoartinerf']}
  • Theorem 4.1: GWi2
  • ...and 4 more