Two-generator Discrete Subgroups of Isom(H2) Containing Orientation-reversing Elements

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Let f and g be elements of the isometry group Isom(H2) of the hyperbolic plane H2, and assume that one of them is orientation-reversing. We determine when the group 〈f,g〉 they generate is discrete; in particular, we obtain the classification of such groups. As an application to knot theory, we completely determine the tunnel number one Montesinos knots.

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