Singular sets and parameters of generalized triangle orbifolds

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a method to obtain the parameters defining a generalized triangle group from the structure of the singular set of its quotient orbifold and illustrate it by examples.

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