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The geometry of the Eisenstein-Picard modular group

Falbel, Elisha; Parker, John R

The geometry of the Eisenstein-Picard modular group Thumbnail


Elisha Falbel


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 ${\rm PU}(2,1)$


Falbel, E., & Parker, J. R. (2006). The geometry of the Eisenstein-Picard modular group. Duke Mathematical Journal, 131(2), 249-289.

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
Publisher URL


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