  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.
