Generators of the Eisenstein–picard Modular Group

Abstract

We prove that the Eisenstein-Picard modular group SU(2, 1; Z[ω 3 ]) can be generated by four given transformations.2010 Mathematics subject classification: primary 32M05; secondary 22E40, 32M15.

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

“…This algorithm was extended to PU(2, 1; Z[i]) by Falbel et al [3] thus giving a different system of generators from that obtained via a fundamental domain in [2]. In [11], the authors applied the continued fraction algorithm to PU(2, 1; Z[ω]) and so produce a different generating system from that obtained in [7].…”

Generators of the sister of Euclidean Picard modular groups

“…In this paper, we will only consider the case d = 2, 7, 11. Let Γ (d) s be the collection of all elements of PU(2, 1) that, when written in the form (3), have z 11 , z 12 , z 13 It is simple to check that Γ (d) s is a group. It is also discrete as O d is discrete in C. We will show that the sister groups Γ and Γ (d) .…”

“…In particular, we do not need to know the fundamental domain for the action of lattice on the complex hyperbolic plane. See the works [5,13,14]. However, it is hard to get more information of the lattices by this method.…”

Untitled

“…)ޚ‬ This algorithm was extended to PU(2, 1; ᏻ 1 ) in [Falbel et al 2011a], which provided a different system of generators from those obtained via a fundamental domain in [Falbel et al 2011b]. In [Wang et al 2011], the authors applied the continued fraction algorithm to PU(2, 1; ᏻ 3 ) and produced a different system of generators from that obtained in [Falbel and Parker 2006].…”