My team and I studied the hypergeometric group in GL_3(CC) with

parameters α=(1/4,1/2,3/4) and β=(0,0,0). In this talk I give a new proof

that this group is isomorphic to the free product Z/4Z∗Z/2Z by exhibiting

a ping-pong table. This table is determined by a simplicial cone in R^3,

and I will prove that this is the unique simplicial cone (up to sign) for

which this construction produces a valid ping-pong table. This talk is

based on joint work with Gabriel Frieden and Étienne Soucis.