Cactus groups, twin groups, and right-angled Artin groups

2022/09/19 by Paolo Bellingeri, Bellingeri, Paolo, Hugo Chemin +3 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2209.08813

openalex publication_date 2022/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Cactus groups Jn are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups, and in particular twin groups Twn and Mostovoy's Gauss diagram groups Dn, which are better understood. Concretely, we construct an injective group 1-cocycle from Jn to Dn, and show that Twn (and its k-leaf generalisations) inject into Jn. As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, PJn. In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group PJ4. Our tools come mainly from combinatorial group theory.

Cited by

• Twin groups representations

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