Cones and ping-pong in three dimensions

Abstract

We study the hypergeometric group in ${\mathrm{GL}}_3(\mathbb{C})$ 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 $\mathbb{Z}/4\mathbb{Z}\ast \mathbb{Z}/2\mathbb{Z}$ 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.

15 pages, 4 figures

References in corpus (1)

• A cyclotomic family of thin hypergeometric monodromy groups in ${Sp}_4(\mathbb{R})$