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

Paolo Bellingeri (1) , Hugo Chemin (1) , Victoria Lebed (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Keywords: Braid groups; twin groups; cactus groups; right-angled Coxeter groups; pure cactus groups; virtual braid groups; torsion; word problem; normal form; group 1-cocycle.

Domains: Mathematics [math]; Combinatorics [math.CO]

Paolo Bellingeri, Hugo Chemin, Victoria Lebed. Cactus groups, twin groups, and right-angled Artin groups. 2022. ⟨hal-03778835v2⟩