Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-31T22:22:43.879Z Has data issue: false hasContentIssue false
  • English
  • Français

A presentation for the Eisenstein-Picard modular group in three complex dimensions

Part of: Special aspects of infinite or finite groups Other groups of matrices Lie groups Real and complex geometry

Published online by Cambridge University Press:  25 July 2023

Jieyan Wang  and
Baohua Xie Open the ORCID record for Baohua Xie [Opens in a new window]
Jieyan Wang
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, P.R. China
Baohua Xie*
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, P.R. China
*
Corresponding author: Baohua Xie; Email: xiexbh@hnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

A. Mark and J. Paupert [Presentations for cusped arithmetic hyperbolic lattices, 2018, arXiv:1709.06691.] presented a method to compute a presentation for any cusped complex hyperbolic lattice. In this note, we will use their method to give a presentation for the Eisenstein-Picard modular group in three complex dimensions.

Keywords

Complex hyperbolic spaceEisenstein-Picard modular groupCoarse fundamental domain

MSC classification

Primary: 22E40: Discrete subgroups of Lie groups
Secondary: 20F05: Generators, relations, and presentations 20H10: Fuchsian groups and their generalizations 51M10: Hyperbolic and elliptic geometries (general) and generalizations

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 3 , September 2023 , pp. 625 - 654
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Cohn, P. M., On the structure of the $GL_2$ of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5–53.CrossRefGoogle Scholar
Cohn, P. M., A presentation of $SL_2$ for Euclidean imaginary quadratic number fields, Mathematika 15 (1968), 156–163.Google Scholar
Falbel, E., Francsics, G., Lax, P. D. and Parker, J. R., Generators of a Picard modular group in two complex dimensions, Proc. Am. Math. Soc. 139(7) (2011), 2439–2447.Google Scholar
Falbel, E., Francsics, G. and Parker, J. R., The geometry of the Gauss-Picard modular group, Math. Ann. 349(2) (2011), 459–508.Google Scholar
Falbel, E. and Parker, J. R., The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131(2) (2006), 249–289.Google Scholar
Goldman, W. M., Complex hyperbolic geometry, Oxford Mathematical Monographs (Oxford University Press, Oxford, New York, 1999).Google Scholar
Macbeath, A. M., Groups of homeomorphisms of a simply connected space, Ann. Math. 79 (1964), 473–488.CrossRefGoogle Scholar
Mark, A. and Paupert, J., Presentations for cusped arithmetic hyperbolic lattices, 2018, arXiv:1709.06691.Google Scholar
Parker, J. R., Notes on complex hyperbolic geometry (2015). Available at: https://www.maths.dur.ac.uk/users/j.r.parker/ img/NCHG.pdf.Google Scholar
Parker, J. R., On the volumes of cusped, complex hyperbolic manifolds and orbifolds, Duke Math. J. 94(3) (1998), 433–464.CrossRefGoogle Scholar
Polletta, D., Presentations for the Euclidean Picard modular groups, Geom. Dedicata 210(1) (2021), 1–26.Google Scholar
Stover, M., Volumes of Picard modular surfaces, Proc. Am. Math. Soc. 139(9) (2011), 3045–3056.Google Scholar
Swan, R. G., Generators and relations for certain special linear groups, Adv. Math. 6(1) (1971), 1–77.Google Scholar
Wang, J., Xiao, Y. and Xie, B., Generators of the Eisenstein-Picard modular group, J. Aust. Math. Soc. 91(3) (2011), 421–429.Google Scholar
Wang, J. and Xie, B., Generators and relations of the Eisenstein-Picard modular group in three complex dimensions. 2022. Available at: https://github.com/Jieyanwang/Presentation-Eisenstein-Picard-Modular-Group.git.Google Scholar
Xie, B., Generators of the sister of Euclidean Picard modular groups, Math. Z. 286(1-2) (2017), 521–543.Google Scholar
Xie, B., Wang, J. and Jiang, Y., Generators of the Eisenstein-Picard modular group in three complex dimensions, Glasg. Math. J. 55(3) (2013), 645–654.Google Scholar
Xie, B., Wang, J. and Jiang, Y., Generators of the Gauss-Picard modular group in three complex dimensions, Pacific J. Math. 273(1) (2015), 197–211.Google Scholar
Zhao, T., A minimal volume arithmetic cusped complex hyperbolic orbifold, Math. Proc. Cambridge Philos. Soc. 150(2) (2011), 313–342.Google Scholar
Zhao, T., Generators for the Euclidean Picard modular groups, Trans. Am. Math. Soc. 364(6) (2012), 3241–3263.Google Scholar