Rank 1 character varieties of finitely presented groups

Abstract

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.

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 Dedicata

References in corpus (4)

• Minimal Affine Coordinates for SL(3,C) Character Varieties of Free Groups
• Rank 1 character varieties of finitely presented groups
• Varieties of Characters
• Invariants of pairs in SL(4,C) and SU(3,1)

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• Kauffman bracket skein algebra of the 4-holed disk
• The $\mathrm{SL}(2,\mathbb{C})$-character variety of the Borromean link