Generators of the Eisenstein–picard Modular Group in Three Complex Dimensions

Abstract

Little is known about the generators system of the higher dimensional Picard modular groups. In this paper, we prove that the higher dimensional EisensteinPicard modular group PU(3, 1; ‫[ޚ‬ω 3 ]) in three complex dimensions can be generated by four given transformations.2000 Mathematics Subject Classification. Primary 32M05, 22E40; Secondary 32M15. Introduction. As the complex hyperbolic analogue of Bianchi groupsPSL(2; O d ), Picard modular groups are PU(n, 1; O d ), where O d is the ring of algebraic integers of the imaginary quadratic extension Q(i √ d) for any positive square-free integer d (see [9]). The elements of the ring O d can be described (see [8]):

Generators of the Gauss–Picard modular group in three complex dimensions

“…explicit fundamental domain, generators system of the higher dimensions Picard modular groups PU(n, 1; O d ). In [12], the continued fraction algorithm had been generalised to Picard modular groups in higher complex dimensions. It contained the first generalisation that we were aware of to a group of 4 × 4 matrices.…”

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“…That is, ∞ ≡ {g ∈ PU(n, 1) : g(q ∞ ) = q ∞ }. We recall from [Falbel et al 2011a;Francsics and Lax 2005a;2005b;Xie et al 2013] that the Langlands decomposition can be used to parametrize a transformation in the stabilizer subgroup of q ∞ .…”

“…We observe that very little is known about the geometry and algebraic properties, e.g., explicit fundamental domain or generating system of the higherdimensional Picard modular groups PU(n, 1; ᏻ d ). In [Xie et al 2013], the continued fraction algorithm was generalized to Picard modular groups in higher complex dimensions. It contained the first generalization that we were aware of to a group of 4 × 4 matrices.…”

“…However, it seems very difficult to extend the continued fraction algorithm to other higher-dimensional Picard modular groups. Using a combination of the ideas from [Falbel et al 2011a;Xie et al 2013] and [Falbel and Parker 2006;Zhao 2012], we will present a method to obtain the generating system of the Gauss-Picard modular group PU(3, 1; ‫[ޚ‬i ]). We first get the generators of the stabilizer of infinity of PU(3, 1; ‫[ޚ‬i ]) by applying a similar argument as in our previous paper [Xie et al 2013].…”

“…Using a combination of the ideas from [Falbel et al 2011a;Xie et al 2013] and [Falbel and Parker 2006;Zhao 2012], we will present a method to obtain the generating system of the Gauss-Picard modular group PU(3, 1; ‫[ޚ‬i ]). We first get the generators of the stabilizer of infinity of PU(3, 1; ‫[ޚ‬i ]) by applying a similar argument as in our previous paper [Xie et al 2013]. Then we will construct a subset in the boundary of complex hyperbolic space which contains the fundamental domain for the stabilizers of infinity in PU (3, 1; ‫[ޚ‬i ]).…”