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Free Products and the Isomorphism between Standard and Dual Artin Groups

Sirio Resteghini

Abstract

Given a Coxeter system with a fixed Coxeter element, there is a surjective group morphism $Ψ$ from the standard to the dual Artin groups. We give conditions that are sufficient, necessary or equivalent to $Ψ$ being an isomorphism. In particular, we prove that if the Hurwitz action on the reduced words of any element in the noncrossing partition poset is transitive, and if the Hurwitz action on the reduced words of the Coxeter element has the same stabilizer as essentially the same action viewed in the standard Artin group, then $Ψ$ is an isomorphism. Both of those conditions are already known in some cases, notably in spherical and affine types. We then prove that taking the free (or direct) product of groups that satisfy those two conditions yields another group that, with a suitable Coxeter element, also satisfies them.

Free Products and the Isomorphism between Standard and Dual Artin Groups

Abstract

Given a Coxeter system with a fixed Coxeter element, there is a surjective group morphism from the standard to the dual Artin groups. We give conditions that are sufficient, necessary or equivalent to being an isomorphism. In particular, we prove that if the Hurwitz action on the reduced words of any element in the noncrossing partition poset is transitive, and if the Hurwitz action on the reduced words of the Coxeter element has the same stabilizer as essentially the same action viewed in the standard Artin group, then is an isomorphism. Both of those conditions are already known in some cases, notably in spherical and affine types. We then prove that taking the free (or direct) product of groups that satisfy those two conditions yields another group that, with a suitable Coxeter element, also satisfies them.

Paper Structure

This paper contains 25 sections, 40 theorems, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Techniques used
  4. Structure of this paper
  5. Acknowledgements
  6. Background
  7. Coxeter and Artin groups
  8. Reduced words and lengths
  9. The dual Artin group
  10. The Hurwitz action
  11. The case of free groups
  12. A morphism from the standard Artin group to the dual Artin group
  13. $[1,g]_R$ projects onto $[1,h]_T$
  14. The Hurwitz presentation for $\mathop{\mathrm{Art}}\nolimits^*(W,T,h)$
  15. Definition of the morphism $\Psi$
  16. ...and 10 more sections

Key Result

Theorem 2.1

The Hurwitz action on $\mathop{\mathrm{Red}}\nolimits_T(h)$ is transitive. (igusa2009exceptional, see also transhurwitz)

Figures (5)

  • Figure 1: Loops representing $f_1,f_2,\dots,f_n,g$. The loops are followed counterclockwise.
  • Figure 2: The black circle is the image of a noncrossing loop that represents $g$. Let $F:\mathbb{C}\times[0,1]\rightarrow \mathbb{C}$ be an isotopy such that $F(\cdot,0)$ is the identity, the restriction of $F(\cdot,t)$ outside of the disk delimited by the red circle is the identity for each $t$, and the blue paths are $F(x_i,\cdot)$ and $F(x_{i+1},\cdot)$. The homeomorphism $\alpha_i$ is the restriction of $F(\cdot,1)$ to $\mathbb{C}\setminus\{x_1,\dots,x_n\}$.
  • Figure 3: We denote by $R_1,R_2$ the regions of $\mathbb{C}$ delimited by the image of $\gamma_g$ and by the red curves, containing $x_k, x_{k+1}$ respectively.
  • Figure 4: Example of construction of $\gamma_1,\dots,\gamma_{\tilde{N}}$. First, intersect the image of $\gamma$ with $R_1\cup R_2$. Then, complete each of the connected components of this intersection to a loop based in $x_0$, maintaining their orientation and in such a way that they do not self-intersect or intersect each other (except in $x_0$). Finally, number these loops such that $[\gamma]=[\gamma_1]\cdot[\gamma_2]\cdot\ldots\cdot[\gamma_{\tilde{N}}]$
  • Figure 5: $\gamma_g$ is a noncrossing path representing $g$ such that $\gamma\subseteq\gamma_g$

Theorems & Definitions (79)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.4
  • proof
  • ...and 69 more