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

BaoHua Xie, JieYan Wang and YuePing Jiang

Vol. 273 (2015), No. 1, 197–211

DOI: 10.2140/pjm.2015.273.197

Abstract

In this paper, we will describe a method to obtain the generators system of Gauss–Picard modular group $PU\left(3,1;\mathbb{Z}\left[i\right]\right)$. More precisely, we will show that $PU\left(3,1;\mathbb{Z}\left[i\right]\right)$ can be generated by five given transformations: two Heisenberg translations, two Heisenberg rotations and one involution. Indeed, the same method works for the other higher-dimensional Euclidean Picard modular groups.

Keywords

complex hyperbolic space, Picard modular groups, generators

Mathematical Subject Classification 2010

Primary: 32M05, 22E40

Secondary: 32M15

Milestones

Received: 11 January 2014

Revised: 27 April 2014

Accepted: 1 June 2014

Published: 6 December 2014