Generators for the Euclidean Picard modular groups

Zhao, Tiehong

doi:10.1090/S0002-9947-2011-05524-8

Generators for the Euclidean Picard modular groups

Author: Tiehong Zhao
Journal: Trans. Amer. Math. Soc. 364 (2012), 3241-3263
MSC (2010): Primary 22E40; Secondary 32M15, 32Q45
DOI: https://doi.org/10.1090/S0002-9947-2011-05524-8
Published electronically: November 7, 2011
MathSciNet review: 2888244
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Abstract: The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain a presentation of the isotropy subgroup fixing infinity by analysis of the combinatorics of the fundamental domain in the Heisenberg group.

1. P. M. Cohn, A presentation of 𝑆𝐿₂ for Euclidean imaginary quadratic number fields, Mathematika 15 (1968), 156–163. MR 0236271, https://doi.org/10.1112/S0025579300002515
2. Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. MR 1482520
3. 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, https://doi.org/10.1007/BF02393220
4. Jan-Michael Feustel, Zur groben Klassifikation der Picardschen Modulflächen, Math. Nachr. 118 (1984), 215–251 (German). MR 773623, https://doi.org/10.1002/mana.19841180117
5. E. Falbel, G. Francsics, P. Lax and J. R. Parker. Generators of a Picard modular group in two complex dimensions. Proc. Amer. Math. 139 (2011), 2439-2447.
6. J.-M. Feustel and R.-P. Holzapfel, Symmetry points and Chern invariants of Picard modular surfaces, Math. Nachr. 111 (1983), 7–40. MR 725771, https://doi.org/10.1002/mana.19831110102
7. Benjamin Fine, Algebraic theory of the Bianchi groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 129, Marcel Dekker, Inc., New York, 1989. MR 1010229
8. 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, https://doi.org/10.1215/S0012-7094-06-13123-X
9. Elisha Falbel, Gábor Francsics, and John R. Parker, The geometry of the Gauss-Picard modular group, Math. Ann. 349 (2011), no. 2, 459–508. MR 2753829, https://doi.org/10.1007/s00208-010-0515-5
10. 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, https://doi.org/10.1090/conm/368/06780
11. G. Francsics and P. Lax. An explicit fundamental domain for the Picard Modular Group in two complex dimensions. (Preprint 2005).
12. William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
13. 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 0267041, https://doi.org/10.2307/1970838
14. Rolf-Peter Holzapfel, Ball and surface arithmetics, Aspects of Mathematics, E29, Friedr. Vieweg & Sohn, Braunschweig, 1998. MR 1685419
15. G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
16. John R. Parker, Cone metrics on the sphere and Livné’s lattices, Acta Math. 196 (2006), no. 1, 1–64. MR 2237205, https://doi.org/10.1007/s11511-006-0001-9
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. MR 2581913, https://doi.org/10.1090/conm/501/09838
18. Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, https://doi.org/10.1112/blms/15.5.401
19. 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. MR 549770
20. Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1–77 (1971). MR 0284516, https://doi.org/10.1016/0001-8708(71)90027-2
21. Thomas Zink, Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen, Math. Nachr. 89 (1979), 315–320 (German). MR 546890, https://doi.org/10.1002/mana.19790890125
22. Tiehong Zhao, A minimal volume arithmetic cusped complex hyperbolic orbifold, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 2, 313–342. MR 2770066, https://doi.org/10.1017/S0305004110000526