  We give an arithmetic criterion which is sufficient to imply the discreteness
of various two-generator subgroups of $PSL(2,{\bold C})$. 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.

Differential Geometry