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

Cactus groups \(J_n\) are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups \(Tw_n\) and Mostovoy’s Gauss diagram groups \(D_n\), which are better understood. Concretely, we construct an injective group 1-cocycle from \(J_n\) to \(D_n\) and show that \(Tw_n\) (and its k-leaf generalizations) inject into \(J_n\). 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, \(PJ_n\). In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group \(PJ_4\). Our tools come mainly from combinatorial group theory.