A discreteness algorithm for 4-punctured sphere groups

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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.