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.
CITATION STYLE
Gehring, F. W., Maclachlan, C., Martin, G. J., & Reid, A. W. (1997). Arithmeticity, discreteness and volume. Transactions of the American Mathematical Society, 349(9), 3611–3643. https://doi.org/10.1090/s0002-9947-97-01989-2
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