A presentation of the pure cactus group of degree four

Abstract

We give a simple presentation of the pure cactus group $PJ_4$ of degree four. This presentation is obtained by considering an action of $PJ_4$ on the hyperbolic plane and constructing a Dirichlet polygon for the action. As a corollary, we provide a direct alternative proof that $PJ_4$ is isomorphic to the fundamental group of the connected sum of five real projective planes.