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Combinatorics of affine cactus groups

Hugo Chemin

TL;DR

The paper shows that affine cactus groups $AJ_n$ are precisely the generalized cactus groups on the affine symmetric group $W(\widetilde{A}_n)$, yielding $AJ_n\cong C_{W(\widetilde{A}_n)}$. It provides a robust embedding $AJ_n\hookrightarrow AD_n \rtimes S_n$, giving linearity and enabling a solvable word problem; it also establishes residual nilpotence for the pure affine cactus groups $PAJ_n$, and derives a semi-direct decomposition $AJ_n\simeq \langle\langle AJ_n^{2,p-1}\rangle\rangle \rtimes AJ_n^{p,n}$. The study clarifies the torsion structure (no odd-order torsion; even-order torsion with $2^k$-order elements for large $n$) and situates $J_n$ as a subgroup, linking to Twin groups and Gauss diagram groups. Overall, the work provides a cohesive combinatorial and algebraic framework for affine cactus groups with algorithmic and structural consequences.

Abstract

This article deals with the study of affine cactus groups from a combinatorial point of view. Those groups are extensions of cactus groups, which are related to braid and diagram groups and have gained an important place in many mathematics topics. We first show that affine cactus groups may be described as cactus groups on Coxeter groups of type eAn. Then, we prove that these groups embed into a semi-direct product of Coxeter groups, which allows us to obtain a number of combinatorial properties of affine cactus groups, such as the solubility of the world problem or the fact that their centre is trivial.

Combinatorics of affine cactus groups

TL;DR

The paper shows that affine cactus groups are precisely the generalized cactus groups on the affine symmetric group , yielding . It provides a robust embedding , giving linearity and enabling a solvable word problem; it also establishes residual nilpotence for the pure affine cactus groups , and derives a semi-direct decomposition . The study clarifies the torsion structure (no odd-order torsion; even-order torsion with -order elements for large ) and situates as a subgroup, linking to Twin groups and Gauss diagram groups. Overall, the work provides a cohesive combinatorial and algebraic framework for affine cactus groups with algorithmic and structural consequences.

Abstract

This article deals with the study of affine cactus groups from a combinatorial point of view. Those groups are extensions of cactus groups, which are related to braid and diagram groups and have gained an important place in many mathematics topics. We first show that affine cactus groups may be described as cactus groups on Coxeter groups of type eAn. Then, we prove that these groups embed into a semi-direct product of Coxeter groups, which allows us to obtain a number of combinatorial properties of affine cactus groups, such as the solubility of the world problem or the fact that their centre is trivial.

Paper Structure

This paper contains 7 sections, 25 theorems, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Affine cactus groups are generalized cactus groups
  3. Linearity and residual nilpotence
  4. Direct consequences
  5. Word problem and notable subgroups
  6. Relations between usual cactus groups and affine cactus groups
  7. Torsion in affine cactus groups

Key Result

Theorem A

For all $n \geq 2$, we have

Figures (5)

  • Figure 1: Diagramatic representation of the affine cactus $\sigma_{1,2}\sigma_{3,1}\sigma_{2,3}$ dans $AJ_4$
  • Figure 2: Examples of relations in affine cactus groups
  • Figure 3: Coxeter diagram of type $\widetilde{A}$
  • Figure 4: Action of $\omega_{\{2,3,4,6,7\}}$ on the graph of a Coxeter system $(W,\{1, \ldots, 10\})$
  • Figure 5: Coxeter diagram of $(W_I,I)$

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Proposition 2.2: bourbaki_groupes_nodate
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: bourbaki_groupes_nodate
  • Definition 2.6: ilin_moduli_2023,bonnafe_cells_2016
  • ...and 44 more