Volumes and arithmeticity of π/3-equiangular hyperbolic polyhedra

Abstract

A hyperbolic polyhedron is called π/3-equiangular if all its dihedral angles are equal to π/3. We find a sequence {P_n} of π/3-equiangular polyhedra different from the sequence Atkinson found in [4]. Atkinson [4] showed that ideal regular tetrahedron P₁ has the smallest volume among all π/3-equiangular hyperbolic polyhedra. In this paper, we show that ideal regular cube P₂ has the second smallest volume and pentagonal prism has the third smallest volume among π/3-equiangular polyhedra. Moreover, we have shown that the reflection groups obtained from {P_n} are arithmetic and that of pentagonal prism is non-arithmetic.