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Let X(F,G) be the G-character variety of F where G is a rank 1 complex affinealgebraic group and F is a finitely presentable discrete group. We describe analgorithm, which we implement in Mathematica, SageMath, and in Python, thattakes a finite presentation for F and produces a finite presentation of thecoordinate ring of X(F,G). We also provide a new description of the definingrelations and local parameters of the coordinate ring when F is free. Althoughthe theorems used to create the algorithm are not new, we hope that as awell-referenced exposition with a companion computer program it will be usefulfor computation and experimentation with these moduli spaces.

Rank 1 character varieties of finitely presented groups