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

TL;DR

This paper investigates the algebraic structure of cactus groups by establishing their relations to well-understood groups like twin and Gauss diagram groups, solving fundamental problems such as the word problem and torsion properties.

Contribution

It constructs an injective group 1-cocycle from cactus groups to Gauss diagram groups and embeds twin groups into cactus groups, providing new insights and solutions to classical problems.

Findings

Solved the word problem for cactus groups

Determined that cactus groups have only even torsion

Established the triviality of the center of cactus groups

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.