ARITHMETICITY, DISCRETENESS AND VOLUME

We give an arithmetic criterion which is sufficient to imply the discr eteness of various two-generator subgroups of PSL(2, c). We then exami ne certain two-generator groups which arise as extremals in various ge ometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margu lis constant and the minimal distance between elliptic axes. We establ ish the discreteness and arithmeticity of a number of these extremal g roups, the associated minimal volume arithmetic group in the commensur ability class and we study whether or not the axis of a generator is s imple. We then list all ''small'' discrete groups generated by ellipti cs of order 2 and n, n = 3, 4, 5, 6, 7.