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Generators of a Picard modular group in two complex dimensions. (English) Zbl 1222.32038

Denote by \(\text{SU}(2,1)\) the subgroup of \(\text{SL}(3,{\mathbb C})\) leaving invariant the quadratic form
\[ C(\zeta)=-\zeta_2\bar \zeta_0-\zeta_0\bar \zeta_2+|\zeta_1|^2\quad\text{with}\quad \zeta=(\zeta_0,\zeta_1,\zeta_2)\in{\mathbb C}^3. \]
Then \(\text{SU}(2,1)\) acts by rational holomorphic automorphisms on the hermitian symmetric space
\[ D^2=\bigg\{z=(z_1,z_2)\in{\mathbb C}^2\;\Big|\; \text{Re}(z_2)>{1\over 2}|z_1|^2\bigg\}. \]
The Picard modular groups are the groups \(\text{SU}(2,1;{\mathcal O}_d)\), where \({\mathcal O}_d\) is the ring of algebraic integers of an imaginary quadratic extension \({\mathbb Q}\big(i\sqrt d\big)\), with \(d\) a positive squarefree integer. Hence they are discrete holomorphic automorphism subgroups of \(D^2\).
In this paper, it is proved that the Picard modular group \(\text{SU}(2,1;{\mathcal O}_d)\) obtained for \(d=1\) is generated by two Heisenberg translations, a rotation, and an involution, namely by the elements
\[ N_{(0,i)}\equiv\begin{pmatrix} 1&0&0\cr 0&1&0\cr i&0&1,\end{pmatrix},\quad N_{(1+i,1)}\equiv\begin{pmatrix} 1&0&0\cr 1+i&1&0\cr 1&1-i&1,\end{pmatrix},\quad M_i\equiv \begin{pmatrix} i&0&0\cr 0&-1&0\cr 0&0&i\end{pmatrix}, \] and
\[ J\equiv \begin{pmatrix} 0&0&-1\cr 0&-1&0\cr -1&0&0 \end{pmatrix}. \]
The method of proof is constructive: it gives an algorithm to decompose any transformation in \(\text{SU}(2,1;{\mathcal O}_d)\) as a product of the four above generators. An important ingredient is the fact that, for \(d=1\), the ring \({\mathcal O}_d \) is a Euclidean ring.
In [G. Francsics and P. D. Lax, Geometric analysis of PDE and several complex variables. Dedicated to François Trèves. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 368, 211–226 (2005; Zbl 1065.22007)] and in [G. Francsics and P. D. Lax, “An explicit fundamental domain for the Picard modular group in two complex dimensions”, arXiv:math/0509708], a fundamental domain for the Picard modular group \(\text{SU}(2,1;{\mathcal O}_d)\) is explicitly constructed; in [G. Francsics and P. D. Lax, Proc. Nat. Acad. Sciences USA 103, No. 30, 11103–11105 (2006; Zbl 1206.11048)], some applications to the Maass cusp forms of the automorphic Laplace-Beltrami operator of \(\text{SU}(2,1;{\mathcal O}_d)\) are given.

MSC:

32M05 Complex Lie groups, group actions on complex spaces
22E40 Discrete subgroups of Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:

[1] Ling Bao, Axel Kleinschmidt, Bengt E. W. Nilsson, Daniel Persson, and Boris Pioline, Instanton corrections to the universal hypermultiplet and automorphic forms on \?\?(2,1), Commun. Number Theory Phys. 4 (2010), no. 1, 187 – 266. · Zbl 1209.81160 · doi:10.4310/CNTP.2010.v4.n1.a5
[2] Martin Deraux, Elisha Falbel, and Julien Paupert, New constructions of fundamental polyhedra in complex hyperbolic space, Acta Math. 194 (2005), no. 2, 155 – 201. · Zbl 1113.22010 · doi:10.1007/BF02393220
[3] 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. · Zbl 0758.32010 · doi:10.1007/BF02392446
[4] 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 TM. · Zbl 1213.14049
[5] 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. · Zbl 1065.22007 · doi:10.1090/conm/368/06780
[6] 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.
[7] Gábor Francsics and Peter D. Lax, Analysis of a Picard modular group, Proc. Natl. Acad. Sci. USA 103 (2006), no. 30, 11103 – 11105. · Zbl 1206.11048 · doi:10.1073/pnas.0603075103
[8] Elisha Falbel and John R. Parker, The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249 – 289. · Zbl 1109.22007 · doi:10.1215/S0012-7094-06-13123-X
[9] William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. · Zbl 0939.32024
[10] William M. Goldman and John R. Parker, Complex hyperbolic ideal triangle groups, J. Reine Angew. Math. 425 (1992), 71 – 86. · Zbl 0739.53055
[11] 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. · Zbl 0206.03603 · doi:10.2307/1970838
[12] R.-P. Holzapfel, Invariants of arithmetic ball quotient surfaces, Math. Nachr. 103 (1981), 117 – 153. · Zbl 0495.14025 · doi:10.1002/mana.19811030109
[13] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. · Zbl 0058.03301
[14] A. Kleinschmidt, D. Persson, e-mail communication, 2008.
[15] Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms, Geom. Funct. Anal. 17 (2007), no. 1, 220 – 251. · Zbl 1137.22011 · doi:10.1007/s00039-006-0589-0
[16] G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171 – 276. · Zbl 0456.22012
[17] John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1 – 42. · Zbl 1200.22004 · doi:10.1090/conm/501/09838
[18] Andrei Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 (1993), no. 1, 79 – 105. · Zbl 0785.11034 · doi:10.1007/BF01895514
[19] 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. · Zbl 1022.53034
[20] Ian Stewart and David Tall, Algebraic number theory, Chapman and Hall, London; A Halsted Press Book, John Wiley & Sons, New York, 1979. Chapman and Hall Mathematics Series. · Zbl 0413.12001
[21] J.M. Woodward, Integral lattices and hyperbolic manifolds, Ph.D. Thesis, University of York, 2006.
[22] Dan Yasaki, An explicit spine for the Picard modular group over the Gaussian integers, J. Number Theory 128 (2008), no. 1, 207 – 234. · Zbl 1225.11066 · doi:10.1016/j.jnt.2007.03.008
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