Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a finite presentation for F and produces a finite presentation of the coordinate ring of X(F,G). We also provide a new description of the defining relations and local parameters of the coordinate ring when F is free. Although the theorems used to create the algorithm are not new, we hope that as a well-referenced exposition with a companion computer program it will be useful for computation and experimentation with these moduli spaces.Comment: 30 pages, Mathematica program at http://math.gmu.edu/~slawton3/trace-identities.nb, SageMath program at http://math.gmu.edu/~slawton3/Main.sagews, Python program at http://math.gmu.edu/~slawton3/charvars.py, accepted for publication at Geometriae Dedicat

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In memoriam: This paper is dedicated to the memory of my beloved friend and mentor, Ralph Brooks Turner, Ph.d. 'Last of the free spirits.'Abstract. Let Γ 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 Γ 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.