Cactus Groups from the Viewpoint of Geometric Group Theory - Genevois, Anthony (Topology Proceedings, 2025)
Cactus groups and their pure subgroups appear in various fields of mathematics and are currently attracting attention from diverse mathematical communities. They share similarities with both right-angled Coxeter groups and braid groups. In this article, our goal is to highlight the tools offered by geometric group theory for the group theoretical study of these groups. Among the new contributions made possible thanks to this geometric perspective, we describe an explicit and efficient solution to the conjugacy problem, and we prove that cactus groups are virtually cocompact special and acylindrically hyperbolic. This has various algebraic consequences. From a purely geometrical point of view, we also prove that cactus groups are pairwise non-quasi-isometric.