Skip to main content
SpringerLink
Log in

The geometry of the Gauss–Picard modular group

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We give a construction of a fundamental domain for \({{\rm PU}(2,1,\mathbb{Z} [i])}\), that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers \({\mathbb{Z} [i]}\). We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Beals M., Fefferman C., Grossman R.: Strictly pseudoconvex domains in C n. Bull. Am. Math. Soc. 8, 125–322 (1983)

    Article  MathSciNet  Google Scholar 

  2. Cohn, L.: The dimension of spaces of automorphic forms on a certain two-dimensional complex domain. Mem. Am. Math. Soc. 1(2), no. 158 (1975)

    Google Scholar 

  3. Cooper, D., Hodgson, C.D., Kerckhoff, S.P.: Three-dimensional Orbifolds and Cone-Manifolds. Math. Soc. Jpn. Mem. 5, (2000)

  4. Deligne P., Mostow G.D.: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. IHES 63, 5–89 (1986)

    MATH  MathSciNet  Google Scholar 

  5. Deligne, P., Mostow, G.D.: Commensurability among Lattices in PU(1, n). In: Annals of Mathematics Studies, vol. 132. Princeton University Press, Princeton (1993)

  6. Deraux M.: Dirichlet domains for the Mostow lattices. Exp. Math. 14(4), 467–490 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Deraux M., Falbel E., Paupert J.: New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math. 194(2), 155–201 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Falbel, E., Francsics, G., Lax, P., Parker, J.R.: Generators of a Picard modular group in two complex dimensions. Proc. Am. Math. Soc. arXiv:0911.1104

  9. Falbel E., Parker J.R.: The geometry of Eisenstein–Picard modular group. Duke Math. J. 131(2), 249–289 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Francsics G., Lax P.: A semi-explicit fundamental domain for the Picard modular group in complex hyperbolic space. Contemp. Math. 368, 211–226 (2005)

    MathSciNet  Google Scholar 

  11. Francsics, G., Lax, P.: An explicit fundamental domain for the Picard modular group in two complex dimensions. arXiv:math/0509708 (2005)

  12. Francsics G., Lax P.: Analysis of a Picard modular group. Proc. Natl. Acad. Sci. USA 103(30), 11103–11105 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Garland H., Raghunathan M.S.: Fundamental domains for lattices in (\({\mathbb{R}}\) -)rank 1 semisimple Lie groups. Ann. Math. (2) 92, 279–326 (1970)

    Article  MathSciNet  Google Scholar 

  14. Goldman W.M.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (1999)

    Google Scholar 

  15. Gromov M.: Volume and bounded cohomology. Publ. Math. IHES 56, 5–99 (1982)

    MATH  MathSciNet  Google Scholar 

  16. Holzapfel R.-P.: Invariants of arithmetic ball quotient surfaces. Math. Nachr. 103, 117–153 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Holzapfel R.-P.: Geometry and Arithmetic Around Euler Partial Differential Equations. Reidel, Dordrecht (1986)

    Google Scholar 

  18. Lehner J.: A Short Course in Automorphic Functions. Holt, Rinehart and Winston, Newy York (1966)

    MATH  Google Scholar 

  19. Mostow G.D.: On a remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math. 86, 171–276 (1980)

    MATH  MathSciNet  Google Scholar 

  20. Parker J.R.: Shimizu’s lemma for complex hyperbolic space. Int. J. Math. 3, 291–308 (1992)

    Article  MATH  Google Scholar 

  21. Parker J.R.: On the volumes of cusped, complex hyperbolic manifolds and orbifolds. Duke Math. J. 94(3), 433–464 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Parker J.R.: Cone metrics on the sphere and Livné’s lattices. Acta Math. 196, 1–64 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Parker J.R.: Complex hyperbolic lattices. Contemp. Math. 501, 1–42 (2009)

    Google Scholar 

  24. Parker J.R., Paupert J.: Unfaithful complex hyperbolic triangle groups II: Higher order reflections. Pac. J. Math. 239, 357–389 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Picard E.: Sur des fonctions de deux variables indépendentes analogues aux fonctions modulaires. Acta Math. 2, 114–135 (1883)

    Article  MathSciNet  Google Scholar 

  26. Picard E.: Sur des formes quadratiques ternaires indéfinies indéterminées conjuguées et sur les fonctions hyperfuchsiennes correspondantes. Acta Math. 5, 121–182 (1884)

    Article  MathSciNet  Google Scholar 

  27. Scott G.P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)

    Article  MATH  Google Scholar 

  28. Series C.: The geometry of the Markoff numbers. Math. Intell. 7, 20–29 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Swan R.G.: Generators and relations for certain special linear groups. Adv. Math. 6, 1–77 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  30. Woodward, J.M.: Integral lattices and hyperbolic manifolds. PhD thesis, University of York (2006)

  31. Yasaki D.: An explicit spine for the Picard modular group over the Gaussian integers. J. Number Theory 128, 207–234 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. Yasaki D.: Elliptic points of the Picard modular group. Monatsh. Math. 156, 391–396 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zink T.: Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen. Math. Nachr. 89, 315–320 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques, Université Pierre et Marie Curie, 75252, Paris, France

    Elisha Falbel

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Gábor Francsics

  3. Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, UK

    John R. Parker

Authors
  1. Elisha Falbel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gábor Francsics
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. John R. Parker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisha Falbel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Falbel, E., Francsics, G. & Parker, J.R. The geometry of the Gauss–Picard modular group. Math. Ann. 349, 459–508 (2011). https://doi.org/10.1007/s00208-010-0515-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0515-5

Mathematics Subject Classification (2000)

Search

Navigation