Arithmeticity, discreteness and volume

Abstract

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of ${PSL}(2,\mathbf {c})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all “small” discrete groups generated by elliptics of order $2$ and $n$, $n=3,4,5,6,7$.