ARITHMETICITY, DISCRETENESS AND VOLUME

F. W. Gehring C. Maclachlan G. J. Martin A. W. Reid

Abstract. We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of P SL(2, 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.

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