Cones and ping-pong in three dimensions

We study the hypergeometric group in GL 3 (C) with parameters α = ( 1 4 , 1 2 , 3 4 ) and β = (0, 0, 0).We give a new proof that this group is isomorphic to the free product Z/4Z * Z/2Z by exhibiting a ping-pong table.Our table is determined by a simplicial cone in 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.