Year: 2026
Type: preprint
Abstract: We classify the finite groups that embed into generalized cactus groups associated with arbitrary Coxeter systems. Every finite subgroup is conjugate to a subgroup of the generalized cactus group associated with a finite standard parabolic subgroup of the underlying Coxeter group. This reduces the classification to explicit lists of finite groups for the finite irreducible Coxeter systems. For types B, D, E, F, and H, we analyze cube stabilizers to obtain these lists. For type $I_2(p)$, we determine the associated generalized cactus group directly from its defining presentation. We explicitly realize every listed group as a subgroup of the corresponding generalized cactus group. When the Coxeter diagram has finitely many vertices, we determine the largest order and all possible orders of finite subgroups. In type A, we prove directly that a finite group embeds into the ordinary cactus group on $n$ strands if and only if it embeds into a Sylow 2-subgroup of the symmetric group on $n$ letters. We also give weighted embedding criteria in types A and B and counterexamples to Genevois's conjecture on finite subgroups of cactus groups.
Source: arXiv (Cornell University)
Author: Kim, Junseok
Institution: Technion – Israel Institute of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223133745