Generators for the Euclidean Picard modular groups. (English) Zbl 1250.22014
Let \({\mathcal K}= \mathbb{Q}(\sqrt{-d})\) be a quadratic imaginary number field. Let \({\mathcal O}_d\) be the ring of algebraic integers of \({\mathcal K}\). In the paper under review the author gives a description of generators for certain Picard modular groups \(PU(2,1;{\mathcal O}_d)\) where the ring \({\mathcal O}_d\) is Euclidean except for \(d= 1,3\). It is shown that these generators are: a rotation, two screw Heisenberg rotations, a vertical translation and an involution.
Moreover the author obtains a presentation of the isotropy subgroup fixing infinity by analysis of the combinatorics of the fundamental domain in the Heisenberg group. The method of the article is to start with finding suitable generators of the stabiliser of infinity of \(PU(2,1;{\mathcal O}_d)\) and then construct a fundamental domain for the stabiliser acting on the boundary of the complex hyperbolic space \(\partial\mathbb{H}^2_{\mathbb{C}}\).
Moreover the author obtains a presentation of the isotropy subgroup fixing infinity by analysis of the combinatorics of the fundamental domain in the Heisenberg group. The method of the article is to start with finding suitable generators of the stabiliser of infinity of \(PU(2,1;{\mathcal O}_d)\) and then construct a fundamental domain for the stabiliser acting on the boundary of the complex hyperbolic space \(\partial\mathbb{H}^2_{\mathbb{C}}\).
Reviewer: Andrzej Szczepański (Gdańsk)
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
| 20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
| 20F38 | Other groups related to topology or analysis |
| 20F05 | Generators, relations, and presentations of groups |
References:
| [1] | P. M. Cohn, A presentation of \?\?\(_{2}\) for Euclidean imaginary quadratic number fields, Mathematika 15 (1968), 156 – 163. · Zbl 0169.34501 · doi:10.1112/S0025579300002515 |
| [2] | Karel Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996. · Zbl 0865.20001 |
| [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. · Zbl 1113.22010 · doi:10.1007/BF02393220 |
| [4] | Jan-Michael Feustel, Zur groben Klassifikation der Picardschen Modulflächen, Math. Nachr. 118 (1984), 215 – 251 (German). · Zbl 0586.14029 · doi: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. · Zbl 1222.32038 |
| [6] | J.-M. Feustel and R.-P. Holzapfel, Symmetry points and Chern invariants of Picard modular surfaces, Math. Nachr. 111 (1983), 7 – 40. · Zbl 0528.14015 · doi: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. · Zbl 0760.20014 |
| [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] | Elisha Falbel, Gábor Francsics, and John R. Parker, The geometry of the Gauss-Picard modular group, Math. Ann. 349 (2011), no. 2, 459 – 508. · Zbl 1213.14049 · doi: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. · Zbl 1065.22007 · doi: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). · Zbl 1065.22007 |
| [12] | William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. · Zbl 0939.32024 |
| [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. · Zbl 0206.03603 · doi:10.2307/1970838 |
| [14] | Rolf-Peter Holzapfel, Ball and surface arithmetics, Aspects of Mathematics, E29, Friedr. Vieweg & Sohn, Braunschweig, 1998. · Zbl 0980.14026 |
| [15] | 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 |
| [16] | John R. Parker, Cone metrics on the sphere and Livné’s lattices, Acta Math. 196 (2006), no. 1, 1 – 64. · Zbl 1100.57017 · doi: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. · Zbl 1200.22004 · doi:10.1090/conm/501/09838 |
| [18] | Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001 · doi: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. · Zbl 0413.12001 |
| [20] | Richard G. Swan, Generators and relations for certain special linear groups, Advances in Math. 6 (1971), 1 – 77 (1971). · Zbl 0221.20060 · doi: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). · Zbl 0424.10020 · doi: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. · Zbl 1254.32040 · doi:10.1017/S0305004110000526 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
