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Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case

Jone Lopez de Gamiz Zearra, Conchita Martínez Pérez

TL;DR

The work extends central RAAG results to a broader Artin-group framework by proving that coherence for Artin groups is equivalent to a free derived subgroup, and by characterizing coherent Artin groups as either direct products with a dihedral factor or free amalgamated products along free abelian subgroups. It further shows that for coherent Artin groups, finitely generated normal subgroups yield virtually abelian quotients unless a direct product decomposition forces containment in a free abelian parabolic factor. In the FC-type, especially for even Artin groups, the authors develop centralizer and quasi-centralizer analyses and apply WPD arguments to establish acylindrical hyperbolicity for many subgroups, extending Minasyan–Osin-type results beyond RAAGs. Collectively, the results connect finiteness properties, parabolic subgroup structure, and geometric group theory in a broad Artin-group setting with potential impact on understanding deformability, subgroup structure, and dynamics inside these groups.

Abstract

We generalize to (certain) Artin groups some results previously known for right-angled Artin groups (RAAGs). First, we generalize a result by Droms, B. Servatius, and H. Servatius, and prove that the derived subgroup of an Artin group is free if and only if the group is coherent. Second, coherent Artin groups over non complete graphs split as free amalgamated products along free abelian subgroups, and we extend to arbitrary Artin groups admitting such a splitting a recent result by Casals-Ruiz and the first author on finitely generated normal subgroups of RAAGs. Finally, we use splittings of even Artin groups of FC-type to generalize results of Minasyan and Osin on acylindrical hyperbolicity of their subgroups.

Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case

TL;DR

The work extends central RAAG results to a broader Artin-group framework by proving that coherence for Artin groups is equivalent to a free derived subgroup, and by characterizing coherent Artin groups as either direct products with a dihedral factor or free amalgamated products along free abelian subgroups. It further shows that for coherent Artin groups, finitely generated normal subgroups yield virtually abelian quotients unless a direct product decomposition forces containment in a free abelian parabolic factor. In the FC-type, especially for even Artin groups, the authors develop centralizer and quasi-centralizer analyses and apply WPD arguments to establish acylindrical hyperbolicity for many subgroups, extending Minasyan–Osin-type results beyond RAAGs. Collectively, the results connect finiteness properties, parabolic subgroup structure, and geometric group theory in a broad Artin-group setting with potential impact on understanding deformability, subgroup structure, and dynamics inside these groups.

Abstract

We generalize to (certain) Artin groups some results previously known for right-angled Artin groups (RAAGs). First, we generalize a result by Droms, B. Servatius, and H. Servatius, and prove that the derived subgroup of an Artin group is free if and only if the group is coherent. Second, coherent Artin groups over non complete graphs split as free amalgamated products along free abelian subgroups, and we extend to arbitrary Artin groups admitting such a splitting a recent result by Casals-Ruiz and the first author on finitely generated normal subgroups of RAAGs. Finally, we use splittings of even Artin groups of FC-type to generalize results of Minasyan and Osin on acylindrical hyperbolicity of their subgroups.
Paper Structure (7 sections, 25 theorems, 56 equations, 1 figure)

This paper contains 7 sections, 25 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Artin groups
  4. Coherent right-angled Artin groups
  5. Coherent Artin groups
  6. Finitely generated normal subgroups of coherent Artin groups
  7. Artin groups of FC-type and parabolic subgroups

Key Result

Theorem 1

Let $\Gamma$ be a labeled finite simplicial graph. The Artin group associated to $\Gamma$, $A_\Gamma$, is coherent if and only if its derived subgroup is free.

Figures (1)

  • Figure 1:

Theorems & Definitions (53)

  • Theorem : Theorem \ref{['Artincoherence_commutator']}
  • Theorem : Theorem \ref{['Artingroup_normal']}
  • Theorem : Theorem \ref{['hyperbolic']}
  • Corollary : Corollary \ref{['acylindrically hyperbolic']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 43 more