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

期刊论文
发表日期：

2024年1月10日

查看全文

Paolo Bellingeri,
Hugo Chemin,
Victoria Lebed

摘要

Cactus groups $J_n$ $J_{n}$ are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups $Tw_n$ $T w_{n}$ and Mostovoy’s Gauss diagram groups $D_n$ $D_{n}$ , which are better understood. Concretely, we construct an injective group 1-cocycle from $J_n$ $J_{n}$ to $D_n$ $D_{n}$ and show that $Tw_n$ $T w_{n}$ (and its k-leaf generalizations) inject into $J_n$ $J_{n}$ . 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, $PJ_n$ $P J_{n}$ . In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group $PJ_4$ $P J_{4}$ . Our tools come mainly from combinatorial group theory.