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Minimal presentation, finite quotients and lower central series of cactus groups

Hugo Chemin, Neha Nanda

Abstract

This article deals with the study of cactus groups from a combinatorial point of view. These groups have been gaining prominence lately in various domains of mathematics, amongst which are their relations with well-known groups such as braid groups, diagram groups, to name a few. We compute a minimal presentation for cactus groups in terms of generators and non-redundant relations. We also construct homomorphisms of these groups onto certain finite groups, which leads to results about finite quotients of cactus groups. More precisely, we prove that all (infinite) dihedral groups appear as quotients of cactus groups. We also investigate the lower central series and its consecutive quotients. While there are already known established similarities with braid groups, we deduce a considerable disparity between the two groups.

Minimal presentation, finite quotients and lower central series of cactus groups

Abstract

This article deals with the study of cactus groups from a combinatorial point of view. These groups have been gaining prominence lately in various domains of mathematics, amongst which are their relations with well-known groups such as braid groups, diagram groups, to name a few. We compute a minimal presentation for cactus groups in terms of generators and non-redundant relations. We also construct homomorphisms of these groups onto certain finite groups, which leads to results about finite quotients of cactus groups. More precisely, we prove that all (infinite) dihedral groups appear as quotients of cactus groups. We also investigate the lower central series and its consecutive quotients. While there are already known established similarities with braid groups, we deduce a considerable disparity between the two groups.
Paper Structure (7 sections, 14 theorems, 93 equations, 3 figures)

This paper contains 7 sections, 14 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A minimal presentation of Cactus groups
  3. From cactus groups to dihedral groups
  4. Lower central series of cactus groups
  5. Computation of $\Gamma_2(J_n)/\Gamma_3(J_n)$
  6. Computation of $\Gamma_3(J_n)/\Gamma_4(J_n)$
  7. Computation of $J_n/\Gamma_3(J_n)$

Key Result

Theorem A

The cactus group $J_n$ is generated by $\{ \sigma_{i} ~|~ i= 2, 3, \dots, n \}$ subject to the following relations: Further, this presentation is minimal in terms of number of generators.

Figures (3)

  • Figure 1: Diagrammatic representation of the element $\sigma_{p,q}$ of $J_n$
  • Figure 2: The cactus $\sigma_{2,3}\sigma_{4,5}\sigma_{1,3}$ of $J_5$
  • Figure 3: Examples of relations in cactus groups

Theorems & Definitions (33)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['Minimal-Presentation-Cactus-Rewritten']}
  • Remark 2.3
  • ...and 23 more