Generators for the Euclidean Picard modular groups
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.NEWLINENEWLINE 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 \(\partial\mathbb{H}^2_{\mathbb{C}}\).
scientific article; zbMATH DE number 6038814
scientific article