Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Splittings and poly-freeness of triangle Artin groups

Xiaolei Wu, Shengkui Ye

TL;DR

This work classifies when triangle Artin groups split as graphs of free groups and establishes virtual poly-freeness in broad settings. It proves Art_{2,3,2M} splits as a graph of free groups if and only if M>5 is even, giving the explicit amalgam Art_{2,3,2M} ≅ F_3 *_{F_7} F_4, and shows Art_{2,3,2M+1} does not admit a nontrivial graph-of-free-groups splitting; finite-type cases with M≤5 have geometric dimension 3, ruling out such splittings. The authors develop a general criterion for poly-freeness of graphs of free groups via algebraically clean decompositions and apply it to multiple HNN extensions, yielding that many triangle Artin groups are virtually poly-free; they also compute commutator-subgroup presentations and prove non-coherence in the pairwise-coprime case by showing Art_{MNP}′ is perfect. Collectively, these results answer several questions of Jankiewicz and Bestvina, and deepen the understanding of the finiteness properties and subgroup structure of triangle Artin groups.

Abstract

We prove that the triangle Artin group $\mathrm{Art}_{23M}$ splits as a graph of free groups if and only if $M$ is greater than $5$ and even. This answers two questions of Jankiewicz \cite[Question 2.2, Question 2.3]{Jan21} in the negative. Combined with the results of Squier and Jankiewicz, this completely determines when a triangle Artin group splits as a graph of free groups. Furthermore, we prove that the triangle Artin groups are virtually poly-free when the labels are not of the form $(2,3, 2k+1)$ with $k\geq 3$. This partially answers a question of Bestvina \cite{Be99}.

Splittings and poly-freeness of triangle Artin groups

TL;DR

This work classifies when triangle Artin groups split as graphs of free groups and establishes virtual poly-freeness in broad settings. It proves Art_{2,3,2M} splits as a graph of free groups if and only if M>5 is even, giving the explicit amalgam Art_{2,3,2M} ≅ F_3 *_{F_7} F_4, and shows Art_{2,3,2M+1} does not admit a nontrivial graph-of-free-groups splitting; finite-type cases with M≤5 have geometric dimension 3, ruling out such splittings. The authors develop a general criterion for poly-freeness of graphs of free groups via algebraically clean decompositions and apply it to multiple HNN extensions, yielding that many triangle Artin groups are virtually poly-free; they also compute commutator-subgroup presentations and prove non-coherence in the pairwise-coprime case by showing Art_{MNP}′ is perfect. Collectively, these results answer several questions of Jankiewicz and Bestvina, and deepen the understanding of the finiteness properties and subgroup structure of triangle Artin groups.

Abstract

We prove that the triangle Artin group splits as a graph of free groups if and only if is greater than and even. This answers two questions of Jankiewicz \cite[Question 2.2, Question 2.3]{Jan21} in the negative. Combined with the results of Squier and Jankiewicz, this completely determines when a triangle Artin group splits as a graph of free groups. Furthermore, we prove that the triangle Artin groups are virtually poly-free when the labels are not of the form with . This partially answers a question of Bestvina \cite{Be99}.
Paper Structure (12 sections, 30 theorems, 68 equations, 4 figures)

This paper contains 12 sections, 30 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic notations and facts
  3. Poly-free groups
  4. Graphs, immersions and oppressive sets
  5. Reidemeister--Schreier rewriting procedure
  6. Splitting of the Artin group $(2,3,2m)$
  7. Non-splitting of the Artin group $(2,3,2m+1)$
  8. Poly-free, algebraically clean and thin
  9. Poly-freeness of triangle Artin groups
  10. Commutator subgroups of triangle Artin groups
  11. Presentations of the commutator subgroups
  12. Coherence of triangle Artin groups

Key Result

Theorem 1.3

When $M$ is odd, the Artin group $\mathrm{Art}_{23M}$ can not split as a nontrivial graph of free groups. When $M>4$ is even, the Artin group $\mathrm{Art}_{23M}$ is isomorphic to $F_{3}\ast _{F_{7}}F_{4},$ an amalgamated product the free groups $F_{3}$ and $F_{4}$ over a subgroup $F_{7}.$

Figures (4)

  • Figure 1: The presentation complex $X_H$
  • Figure 2: Splitting of $A_{2,3,2m}$
  • Figure 3: The action of $A_{I_{2m+1}}$ on a tree $T$, where $x=ab$ acts as a rotation along $q_0$ and $s=(ab)^ma$ acts a reflection along $p_0$.
  • Figure 6: The map $\rho_i: X_B \xrightarrow{=} X_B \rightarrow \bar{X}_B \rightarrow X_A,i=1,2,$ when $P = 2, M \geq 4, N\geq 4$ are even.

Theorems & Definitions (63)

  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 53 more