The geometry of the Eisenstein-Picard modular group

Falbel, Elisha; Parker, John R

doi:10.1215/s0012-7094-06-13123-x

The geometry of the Eisenstein-Picard modular group

Falbel, Elisha; Parker, John R

Authors

Elisha Falbel

Professor John Parker j.r.parker@durham.ac.uk
Professor

Abstract

The Eisenstein-Picard modular group ${\rm PU}(2,1;\mathbb {Z}[\omega])$ is defined to be the subgroup of ${\rm PU}(2,1)$ whose entries lie in the ring $\mathbb {Z}[\omega]$, where $\omega$ is a cube root of unity. This group acts isometrically and properly discontinuously on ${\bf H}^2_\mathbb{C}$, that is, on the unit ball in $\mathbb {C}^2$ with the Bergman metric. We construct a fundamental domain for the action of ${\rm PU}(2,1;\mathbb {Z}[\omega])$ on ${\bf H}^2_\mathbb {C}$, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of ${\rm PU}(2,1)$

Citation

Falbel, E., & Parker, J. R. (2006). The geometry of the Eisenstein-Picard modular group. Duke Mathematical Journal, 131(2), 249-289. https://doi.org/10.1215/s0012-7094-06-13123-x

Journal Article Type	Article
Publication Date	2006-02
Deposit Date	Aug 27, 2008
Publicly Available Date	Aug 27, 2008
Journal	Duke Mathematical Journal
Print ISSN	0012-7094
Publisher	Duke University Press
Peer Reviewed	Peer Reviewed
Volume	131
Issue	2
Pages	249-289
DOI	https://doi.org/10.1215/s0012-7094-06-13123-x
Publisher URL	http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.dmj&issue=1137077882