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Arithmeticity, Discreteness and Volume

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

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.

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F. W. Gehring, C. Maclachlan, G. J. Martin, A. W. Reid. 1995-04-07. Arithmeticity, Discreteness and Volume. https://arxiv.org/abs/math/9504208

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