Generators of a Picard modular group in two complex dimensions
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- by Elisha Falbel, Gábor Francsics, Peter D. Lax and John R. Parker PDF
- Proc. Amer. Math. Soc. 139 (2011), 2439-2447 Request permission
Abstract:
The goal of the article is to prove that four explicitly given transformations, two Heisenberg translations, a rotation and an involution generate the Picard modular group with Gaussian integers acting on the two dimensional complex hyperbolic space. The result answers positively a question raised by A. Kleinschmidt and D. Persson.References
- Ling Bao, Axel Kleinschmidt, Bengt E. W. Nilsson, Daniel Persson, and Boris Pioline, Instanton corrections to the universal hypermultiplet and automorphic forms on $\textrm {SU}(2,1)$, Commun. Number Theory Phys. 4 (2010), no. 1, 187–266. MR 2679380, DOI 10.4310/CNTP.2010.v4.n1.a5
- Martin Deraux, Elisha Falbel, and Julien Paupert, New constructions of fundamental polyhedra in complex hyperbolic space, Acta Math. 194 (2005), no. 2, 155–201. MR 2231340, DOI 10.1007/BF02393220
- C. L. Epstein, R. B. Melrose, and G. A. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), no. 1-2, 1–106. MR 1111745, DOI 10.1007/BF02392446
- E. Falbel, G. Francsics, J.R. Parker, The geometry of the Gauss-Picard modular group, 2009 preprint, pp. 1–38, to appear in Mathematische Annalen. Published online: 4 May 2010, Online First ™.
- Gábor Francsics and Peter D. Lax, A semi-explicit fundamental domain for a Picard modular group in complex hyperbolic space, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 211–226. MR 2126471, DOI 10.1090/conm/368/06780
- G. Francsics, P. Lax, An explicit fundamental domain for the Picard modular group in two complex dimensions, 2005 preprint, pp. 1–25, arXiv:math/0509708.
- Gábor Francsics and Peter D. Lax, Analysis of a Picard modular group, Proc. Natl. Acad. Sci. USA 103 (2006), no. 30, 11103–11105. MR 2242649, DOI 10.1073/pnas.0603075103
- Elisha Falbel and John R. Parker, The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249–289. MR 2219242, DOI 10.1215/S0012-7094-06-13123-X
- William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
- William M. Goldman and John R. Parker, Complex hyperbolic ideal triangle groups, J. Reine Angew. Math. 425 (1992), 71–86. MR 1151314
- H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank $1$ semisimple Lie groups, Ann. of Math. (2) 92 (1970), 279–326. MR 267041, DOI 10.2307/1970838
- R.-P. Holzapfel, Invariants of arithmetic ball quotient surfaces, Math. Nachr. 103 (1981), 117–153. MR 653917, DOI 10.1002/mana.19811030109
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- A. Kleinschmidt, D. Persson, e-mail communication, 2008.
- Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms, Geom. Funct. Anal. 17 (2007), no. 1, 220–251. MR 2306657, DOI 10.1007/s00039-006-0589-0
- G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
- John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1–42. MR 2581913, DOI 10.1090/conm/501/09838
- Andrei Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 (1993), no. 1, 79–105. MR 1204788, DOI 10.1007/BF01895514
- Richard Evan Schwartz, Complex hyperbolic triangle groups, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 339–349. MR 1957045
- Ian Stewart and David Tall, Algebraic number theory, Chapman and Hall Mathematics Series, Chapman and Hall, London; John Wiley & Sons, Inc., New York, 1979. MR 549770
- J.M. Woodward, Integral lattices and hyperbolic manifolds, Ph.D. Thesis, University of York, 2006.
- Dan Yasaki, An explicit spine for the Picard modular group over the Gaussian integers, J. Number Theory 128 (2008), no. 1, 207–234. MR 2382777, DOI 10.1016/j.jnt.2007.03.008
Additional Information
- Elisha Falbel
- Affiliation: Institut de Mathématiques, Université Pierre et Marie Curie, 4 Place Jussieu, Paris, France
- Email: falbel@math.jussieu.fr
- Gábor Francsics
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- Email: francsics@math.msu.edu
- Peter D. Lax
- Affiliation: Courant Institute, New York University, 251 Mercer Street, New York, New York 10012-1185
- Email: lax@courant.nyu.edu
- John R. Parker
- Affiliation: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, United Kingdom
- MR Author ID: 319072
- ORCID: 0000-0003-0513-3980
- Email: j.r.parker@durham.ac.uk
- Received by editor(s): October 8, 2009
- Received by editor(s) in revised form: June 17, 2010
- Published electronically: November 30, 2010
- Additional Notes: The second author is grateful for the hospitality of the Mathematical Sciences Research Institute at Berkeley and the Rényi Mathematical Institute, Budapest.
- Communicated by: Mei-Chi Shaw
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2439-2447
- MSC (2010): Primary 32M05, 22E40; Secondary 32M15
- DOI: https://doi.org/10.1090/S0002-9939-2010-10653-6
- MathSciNet review: 2784810