Skip to main content
Springer Nature Link
Log in

Character varieties for real forms

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

Let \(\varGamma \) be a finitely generated group and G a real form of \(\mathrm {SL}_n(\mathbb {C})\). We propose a definition for the G-character variety of \(\varGamma \) as a subset of the \(\mathrm {SL}_n(\mathbb {C})\)-character variety of \(\varGamma \). We consider two anti-holomorphic involutions of the \(\mathrm {SL}_n(\mathbb {C})\) character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of \(\mathrm {SL}_n(\mathbb {C})\). We study in detail an example: the \(\mathrm {SL}_n(\mathbb {C})\), \(\mathrm {SU}(2,1)\) and \(\mathrm {SU}(3)\) character varieties of the free product \(\mathbb {Z}/{3}\mathbb {Z}*\mathbb {Z}/{3}\mathbb {Z}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Florentino and Lawton give a first proof of this fact in the appendix of [10]; the proof is corrected in Remark 4.7 of [11].

References

  1. Bergeron, M.: The topology of nilpotent representations in reductive groups and their maximal compact subgroups. Geom. Topol. 19(3), 1383–1407 (2015). https://doi.org/10.2140/gt.2015.19.1383

    Article  MathSciNet  MATH  Google Scholar 

  2. Casimiro, A., Florentino, C., Lawton, S., Oliveira, A.: Topology of moduli spaces of free group representations in real reductive groups. Forum Math. 28(2), 275–294 (2014). https://doi.org/10.1515/forum-2014-0049

    Article  MathSciNet  MATH  Google Scholar 

  3. Cooper, D., Culler, M., Gillet, H., Long, D.D., Shalen, P.B.: Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118(1), 47–84 (1994). https://doi.org/10.1007/BF01231526

    Article  MathSciNet  MATH  Google Scholar 

  4. Cooper, D., Long, D.D.: Representation theory and the A-polynomial of a knot. Chaos Solitons Fractals 9(4–5), 749–763 (1998). https://doi.org/10.1016/S0960-0779(97)00102-1. (Knot theory and its applications)

    Article  MathSciNet  MATH  Google Scholar 

  5. Culler, M., Shalen, P.B.: Varieties of group representations and splittings of 3-manifolds. Ann. Math. 117(1), 109 (1983). https://doi.org/10.2307/2006973

    Article  MathSciNet  MATH  Google Scholar 

  6. Deraux, M.: A 1-parameter family of spherical CR uniformizations of the figure eight knot complement. Geom. Topol. 20(6), 3571–3621 (2016). https://doi.org/10.2140/gt.2016.20.3571

    Article  MathSciNet  MATH  Google Scholar 

  7. Deraux, M., Falbel, E.: Complex hyperbolic geometry of the figure eight knot. Geom. Topol. 19(1), 237–293 (2015). https://doi.org/10.2140/gt.2015.19.237

    Article  MathSciNet  MATH  Google Scholar 

  8. Falbel, E.: A spherical CR structure on the complement of the figure eight knot with discrete holonomy. J. Differ. Geom. 79(1), 69–110 (2008). https://doi.org/10.4310/jdg/1207834658

    Article  MathSciNet  MATH  Google Scholar 

  9. Falbel, E., Guilloux, A., Koseleff, P.V., Rouillier, F., Thistlethwaite, M.: Character varieties for SL(3, C): the figure eight knot. Exp. Math. 25(2), 219–235 (2016). https://doi.org/10.1080/10586458.2015.1068249

    Article  MathSciNet  MATH  Google Scholar 

  10. Florentino, C., Lawton, S.: The topology of moduli spaces of free group representations. Math. Ann. 345(2), 453–489 (2009). https://doi.org/10.1007/s00208-009-0362-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Florentino, C., Lawton, S.: Character varieties and moduli of quiver representations. In: In the Tradition of Ahlfors–Bers. VI, Contemporary Mathematics, vol. 590, pp. 9–38. American Mathematical Society, Providence (2013). https://doi.org/10.1090/conm/590/11720

  12. Florentino, C., Lawton, S.: Topology of character varieties of abelian groups. Topol. Appl. 173, 32–58 (2014). https://doi.org/10.1016/j.topol.2014.05.009

    Article  MathSciNet  MATH  Google Scholar 

  13. Goldman, W.M.: Topological components of spaces of representations. Invent. Math. 93(3), 557–607 (1988). https://doi.org/10.1007/BF01410200

    Article  MathSciNet  MATH  Google Scholar 

  14. Goldman, W.M.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. The Clarendon Press, New York (1999)

    Google Scholar 

  15. Goldman, W.M.: The complex-symplectic geometry of SL(2,C)-characters over surfaces. In: Algebraic Groups and Arithmetic, pp. 375–407. Tata Inst. Fund. Res., Mumbai (2004)

  16. Goldman, W.M.: An exposition of results of Fricke (2004). arXiv preprint arXiv:math/0402103

  17. Gongopadhyay, K., Lawton, S.: Invariants of pairs in SL(4, C) and SU(3,1). Proc. Am. Math. Soc. 145(11), 4703–4715 (2017). https://doi.org/10.1090/proc/13638

    Article  MathSciNet  MATH  Google Scholar 

  18. Helgason, S.: Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs, vol. 39, 2nd edn. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  19. Heusener, M.: SL(n,C)-representation spaces of knot groups (2016). arXiv:1602.03825 [math]

  20. Heusener, M., Muñoz, V., Porti, J.: The SL(3, C)-character variety of the figure eight knot. Ill. J. Math. 60(1), 55–98 (2016). https://doi.org/10.1215/ijm/1498032024

    Article  MathSciNet  MATH  Google Scholar 

  21. Lawton, S.: Generators, relations and symmetries in pairs of unimodular matrices. J. Algebra 313(2), 782–801 (2007). https://doi.org/10.1016/j.jalgebra.2007.01.003

    Article  MathSciNet  MATH  Google Scholar 

  22. Lubotzky, A., Magid, A.R.: Varieties of representations of finitely generated groups. Mem. Am. Math. Soc. 58(336), 0–0 (1985). https://doi.org/10.1090/memo/0336

    Article  MathSciNet  MATH  Google Scholar 

  23. Morgan, J.W., Shalen, P.B.: Valuations, trees, and degenerations of hyperbolic structures, I. Ann. Math. 120(3), 401–476 (1984). https://doi.org/10.2307/1971082

    Article  MathSciNet  MATH  Google Scholar 

  24. Parker, J., Will, P.: A complex hyperbolic Riley slice. Geom. Topol. 21(6), 3391–3451 (2017). https://doi.org/10.2140/gt.2017.21.3391

    Article  MathSciNet  MATH  Google Scholar 

  25. Parreau, A.: Espaces de représentations completement réductibles. J. Lond. Math. Soc. 83(3), 545–562 (2011). https://doi.org/10.1112/jlms/jdq076

    Article  MathSciNet  MATH  Google Scholar 

  26. Parreau, A.: Compactification d’espaces de représentations de groupes de type fini. Math. Z. 272(1–2), 51–86 (2012). https://doi.org/10.1007/s00209-011-0921-8

    Article  MathSciNet  MATH  Google Scholar 

  27. Procesi, C.: The invariant theory of n x n matrices. Adv. Math. 19(3), 306–381 (1976). https://doi.org/10.1016/0001-8708(76)90027-X

    Article  MathSciNet  MATH  Google Scholar 

  28. Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81(3), 539–554 (1985). https://doi.org/10.1007/BF01388587

    Article  MathSciNet  MATH  Google Scholar 

  29. Shalen, P.B.: Representations of 3-manifold groups. In: Daverman, R., Sher, R. (eds.) Handbook of Geometric Topology, pp. 955–1044. North-Holland, Amsterdam (2002)

    MATH  Google Scholar 

  30. Sikora, A.: Character varieties. Trans. Am. Math. Soc. 364(10), 5173–5208 (2012). https://doi.org/10.1090/S0002-9947-2012-05448-1

    Article  MathSciNet  MATH  Google Scholar 

  31. Will, P.: Two-generator groups acting on the complex hyperbolic plane. In: Handbook of Teichmüller Theory. Vol. VI, IRMA Lectures in Mathematics and Theoretical Physics, vol. 27, pp. 275–334. European Mathematical Society, Zürich (2016)

Download references

Acknowledgements

The author would like to thank his advisors Antonin Guilloux and Martin Deraux, as well as Pierre Will, Elisha Falbel and Julien Marché for many discussions about this article. He would also like to thank Maxime Wolff, Joan Porti, Michael Heusener, Cyril Demarche and the PhD students of IMJ-PRG for helping him to clarify many points of the paper, as well as the anonymous referee for many improvements of the article.

Author information

Authors and Affiliations

  1. Unité de recherche en Mathématiques, Université du Luxembourg, Maison du Nombre 6, Avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Miguel Acosta

Authors
  1. Miguel Acosta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel Acosta.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was carried out while the author was at IMJ-PRG (UPMC, Paris, France) and IECL (Université de Lorraine, Nancy, France).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Acosta, M. Character varieties for real forms. Geom Dedicata 203, 257–277 (2019). https://doi.org/10.1007/s10711-019-00435-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00435-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation