  Let $\Gamma$ be a subgroup of $PSL(2,R)$ generated by three parabolic
transformations. The main goal of this paper is to present an algorithm to
determine whether or not $\Gamma$ is discrete. Historically discreteness
algorithms have been considered within several broader mathematical paradigms:
the discreteness problem, the construction and deformation of hyperbolic
structures on surfaces and notions of automata for groups. Each of these
approaches yield equivalent results. The second goal of this paper is to give
an exposition of the basic ideas needed to interpret these equivalences,
emphasizing related works and future directions of inquiry.
