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# The geometry of the Eisenstein-Picard modular group.

Falbel, E. and Parker, J. R. (2006) 'The geometry of the Eisenstein-Picard modular group.', Duke mathematical journal., 131 (2). pp. 249-289.

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

Item Type: Article (VoR) Version of Record Download PDF (213Kb) Peer-reviewed http://dx.doi.org/10.1215/S0012-7094-06-13123-X No date available 27 August 2008 February 2006 No date available