Generators for the Euclidean Picard modular groups
type: Item
instance of: scholarly article
title: Generators for the Euclidean Picard modular groups
zbMATH Open document ID: 1250.22014
DOI: 10.1090/S0002-9947-2011-05524-8
author: Tiehong Zhao
published in: Transactions of the American Mathematical Society
publication date: 2012-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.3331
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 6038814
zbMATH Keywords: Picard modular groups
zbMATH Keywords: Bianchi groups
zbMATH Keywords: complex hyperbolic space
MaRDI profile type: Publication
arXiv ID: 1006.3331
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OpenAlex ID: W2963026025
review text: 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 \(\partial\mathbb{H}^2_{\mathbb{C}}\).
reviewed by: Andrzej Szczepański
description: scientific article; zbMATH DE number 6038814