Two-generator Discrete Subgroups of PSL (2, R)
The discreteness problem is the problem of determining whether or not a two-generator subgroup of $PSL(2, R)$ is discrete. Historically, papers on this old and subtle problem have been known for their errors and omissions. This book presents the first complete geometric solution to the discreteness problem by building upon cases previously presented by Gilman and Maskit and by developing a theory of triangle group shinglings/tilings of the hyperbolic plane and a theory explaining why the solution must take the form of an algorithm. This work is a thoroughly readable exposition that captures the beauty of the interplay between the algebra and the geometry of the solution.
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