A presentation of the pure cactus group of degree four

Takatoshi Hama; Kazuhiro Ichihara

paper

A presentation of the pure cactus group of degree four

arXiv:2504.11852

Abstract

We give a simple presentation of the pure cactus group $P J_{4}$ of degree four. This presentation is obtained by considering an action of $P J_{4}$ on the hyperbolic plane and constructing a Dirichlet polygon for the action. As a corollary, we provide a direct alternative proof that $P J_{4}$ is isomorphic to the fundamental group of the connected sum of five real projective planes.

19 pages, 6 figures