On 2-generator subgroups of SO(3)

Abstract

We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.