Cactus groups from the viewpoint of geometric group theory

TL;DR

This paper explores cactus groups using geometric group theory, providing new insights such as an explicit conjugacy problem solution and proving their hyperbolic properties.

Contribution

It introduces a geometric approach to cactus groups, offering an explicit conjugacy problem solution and establishing their hyperbolic and special properties.

Findings

Explicit solution to the conjugacy problem

Cactus groups are virtually cocompact special

Cactus groups are acylindrically hyperbolic

Abstract

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 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.