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Dyer groups: Centres, hyperbolicity, and acylindrical hyperbolicity

Mireille Soergel, Nicolas Vaskou

Abstract

In this article we describe the centres of all Dyer groups. We also give a complete classification of when a Dyer group $D(Γ)$ is hyperbolic or acylindricality hyperbolic, with conditions that can easily be read on the Dyer graph $Γ$.

Dyer groups: Centres, hyperbolicity, and acylindrical hyperbolicity

Abstract

In this article we describe the centres of all Dyer groups. We also give a complete classification of when a Dyer group is hyperbolic or acylindricality hyperbolic, with conditions that can easily be read on the Dyer graph .

Paper Structure

This paper contains 5 sections, 13 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Centres of Dyer groups
  4. Hyperbolicity
  5. Acylindrical hyperbolicity

Key Result

Theorem A

Let $D(\Gamma)$ be an irreducible Dyer group, and suppose that there exists some $v \in V(\Gamma)$ with $f(v) \neq 2$ (in other words, $\Gamma$ does not just define a Coxeter group). Then either $\Gamma$ is a single vertex and $D(\Gamma)$ is cyclic, or $D(\Gamma)$ has trivial centre.

Figures (3)

  • Figure 1: The double edges represent joins between the components $C_i$. Each edge of the join has label $2$. All vertices on the picture have label $2$. The single black edges have label at least $3$.
  • Figure 2: An example of a Dyer graph $\Gamma$ and the associated Dyer graph $\mathop{\mathrm{\widehat{\Gamma}}}\nolimits$ corresponding to the Coxeter group $W(\mathop{\mathrm{\widehat{\Gamma}}}\nolimits)$. The vertices and edges whose label are not specified have label $2$.
  • Figure 3: The irreducible affine Coxeter groups. The graphs for $\widetilde{A_n}$ ($n \geq 2$), $\widetilde{B_n}$ ($n \geq 3$), $\widetilde{C_n}$ ($n \geq 2$) and $\widetilde{D_n}$ ($n \geq 4$) have $n+1$ vertices.

Theorems & Definitions (24)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.7: Theorem A
  • ...and 14 more