A Generating Set for the Picard Modular Group in the Case $$d=11$$ d = 11
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RESEARCH PAPER
A Generating Set for the Picard Modular Group in the Case d = 11 Mahboubeh Ghoshouni1 • Majid Heydarpour1 Received: 1 July 2019 / Accepted: 17 August 2020 / Published online: 18 September 2020 Ó Shiraz University 2020
Abstract In this work, we find a system of generators for the Picard modular group SUð2; 1; O11 Þ. This system contains four transformations, two translations a rotation and an involution. Keywords Complex hyperbolic space Picard modular group Mathematics Subject Classification Primary: 32M05 Secondary: 22E40 32M15.
1 Introduction Let Od be the ring of algebraic integers in the imaginary pffiffiffiffiffiffiffi quadratic number field K ¼ Qð dÞ where d is a positive and square-free integer. According to Hardy and Wright (1954), the elements of the ring Od can be described as follows: pffiffiffi 8 if d 1; 2 ðmod 4Þ < Z½i d pffiffiffi Od ¼ : Z½1 þ i d if d 3 ðmod 4Þ: 2 It is well-known that the ring Od is Euclidean for positive square-free integer d if and only if d ¼ 1; 2; 3; 7; 11; see Stewart and Tall (1979). The subgroups of SU(2, 1) with entries in Od are called Picard modular groups and are denoted by SUð2; 1; Od Þ. They are a natural generalization of Bianchi groups PSL2 ðOd Þ; the simplest arithmetically defined discrete groups. It is interesting to get a system of generators for Picard modular groups. In Falbel et al. (2011) the authors proved that the Picard modular group with Gaussian integers acting on the two-dimensional complex hyperbolic space can be generated by four transformations. Also, in Falbel et al. & Majid Heydarpour [email protected]
(2011) the authors obtained a finite system of generators for SUð2; 1; Z½iÞ in a geometric way. Analogously, generators of the Picard modular group SUð2; 1; Z½xÞ, where x is a cubic root of unity, were studied by Falbel and Parker in Falbel and Parker (2006) and by Wang, Xiao and Xie in Wang et al. (2011). Zhao in Zhao (2012) obtained a system of generators for the Picard modular group pffiffiffiffi SUð2; 1; Z½1þi2 11Þ in a geometric way. In this work we decompose any element of the Picard modular group pffiffiffiffi SUð2; 1; Z½1þi2 11Þ as a product of the generators, two translations a rotation and an involution, in a non-geometric way.
2 Preliminaries In this part, by taking a standard Hermitian form and making a choice of section for C2;1 , we get a model for the complex hyperbolic plane H2C that is a paraboloid in C2 . The definitions can be found in Goldman (1999); Parker (2003). Let C2;1 be the complex vector space of (complex) dimension 3 equipped with a nondegenerate infinite Hermitian form h ; i : C3 C3 ! C of signature (2,1) given by hz; wi ¼ w Cz ¼ z1 w3 þ z2 w2 þ z3 w1 , where z and w are column vectors in C3 , w is the Hermitian transpose of w and C is the nonsingular Hermitian matrix
Mahboubeh Ghoshouni [email protected] 1
Department of Mathematics, University of Zanjan, Zanjan 45371-38791, Iran
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Iran J Sci Technol Trans Sci (2020) 44:1469–1475
0
0 0 B C ¼ @0 1 1 0
1 1 C 0A
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1 z1 B C z ¼ @ z 2 A:
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with two posi
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