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, P J(n). In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group P J(4). Our tools come mainly from combinatorial group theory.