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Rank 1 character varieties of finitely presented groups

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Let \(\mathfrak {X}(\Gamma ,G)\) be the G-character variety of \(\Gamma \) where G is a rank 1 complex affine algebraic group and \(\Gamma \) 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 \(\Gamma \) and produces a finite presentation of the coordinate ring of \(\mathfrak {X}(\Gamma ,G)\). We also provide a new description of the defining relations and local parameters of the coordinate ring when \(\Gamma \) 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.

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Notes

  1. Available at http://math.gmu.edu/~slawton3/trace-identities.nb, http://math.gmu.edu/~slawton3/Main.sagews, and http://math.gmu.edu/~slawton3/charvars.py respectively.

  2. Available at http://math.gmu.edu/~slawton3/trace-identities.nb.

  3. At the time of this writing, our Python program is expected to be incorporated directly into SnapPy.

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Acknowledgements

Lawton thanks Chris Manon and Bill Goldman for helpful conversations. This material is based upon work supported by the National Science Foundation (NSF) under grant number DMS 1321794; the Mathematics Research Communities (MRC) program. All three authors benefited from our time in Snowbird with the MRC (2011 and 2016) and we are very grateful for the wonderful support they provided. In particular, special thanks goes to Christine Stevens. Additionally, Lawton was supported by the Simons Foundation Collaboration grant 245642, and the NSF grant DMS 1309376. Lastly, we acknowledge support from NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties" (the GEAR Network). We thank an anonymous referee for helping improve the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Caleb Ashley

  2. Institut des Hautes Études Scientifiques, Bureau 1E11, IHES, 35 Route de Chartres, Bures-sur-Yvette, 91440, France

    Jean-Philippe Burelle

  3. Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA, 22030, USA

    Sean Lawton

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  1. Caleb Ashley
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Correspondence to Sean Lawton.

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To William Goldman on the occasion of his 60th birthday.

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Ashley, C., Burelle, JP. & Lawton, S. Rank 1 character varieties of finitely presented groups. Geom Dedicata 192, 1–19 (2018). https://doi.org/10.1007/s10711-017-0281-6

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