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Graphical splittings of Artin kernels

Danielle Barquinero, Lorenzo Ruffoni, Kaidi Ye

Abstract

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag-Solitar group of variable rank. In particular for block graphs (e.g. trees), we obtain an explicit rank formula, and discuss some features of the space of fibrations of the associated right-angled Artin group.

Graphical splittings of Artin kernels

Abstract

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag-Solitar group of variable rank. In particular for block graphs (e.g. trees), we obtain an explicit rank formula, and discuss some features of the space of fibrations of the associated right-angled Artin group.

Paper Structure

This paper contains 8 sections, 21 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on right-angled Artin groups
  3. Graphical splittings for RAAGs
  4. Characters and BNS invariant for RAAGs
  5. Splittings of Artin kernels
  6. Splitting dichotomy
  7. Chordal graphs
  8. Block graphs

Key Result

Theorem \oldthetheorem

Let $(\Gamma_1,\Gamma_2,\Gamma_3)$ be a splitting of a connected graph $\Gamma$, let $f:A_{\Gamma} \to \mathbb{Z}$ be non-trivial, and let $f_k$ denote the restriction of $f$ to $A_{\Gamma_k}$ for $k=1,2,3$. Then

Figures (3)

  • Figure 1: Graphs corresponding to $\mathbb{F}_4$, $\mathbb{Z}^4$, and an amalgamated product of two copies of $\mathbb{Z}^3$ over $\mathbb{Z}^2$.
  • Figure 2: In the first three graphs the splitting subgraph is given by the union of the top and the bottom vertices, while in the last one it is given by the union of the left and the right vertices.
  • Figure 3: A non-chordal well-connected graph with a character as in Proposition \ref{['prop:nonchordal']}

Theorems & Definitions (39)

  • Theorem \oldthetheorem
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  • Remark \oldthetheorem
  • Definition \oldthetheorem
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  • Lemma \oldthetheorem
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  • ...and 29 more