Cones and ping-pong in three dimensions

TL;DR

This paper investigates a specific hypergeometric group in three dimensions, providing a new proof of its structure as a free product using a geometric ping-pong argument based on a unique simplicial cone.

Contribution

It offers a novel geometric proof of the group's isomorphism to a free product, utilizing a uniquely determined simplicial cone as a ping-pong table.

Findings

The hypergeometric group is isomorphic to Z/4Z * Z/2Z.

A specific simplicial cone serves as a unique ping-pong table.

The proof introduces a geometric approach to group isomorphism.

Abstract

We study the hypergeometric group in ${\rm GL}_3(\mathbb{C})$ with parameters $\alpha = (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $\beta = (0,0,0)$. We give a new proof that this group is isomorphic to the free product $\mathbb{Z}/4\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $\mathbb{R}^3$, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.