The geometry of the Eisenstein-Picard modular group

Abstract

The Eisenstein-Picard modular group $\mathrm{PU}(2,1;\mathbb{Z}[\omega ])$ is defined to be the subgroup of $\mathrm{PU}(2,1)$ 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 $H_{\mathbb{C}}^2$, that is, on the unit ball in ${\mathbb{C}}^2$ with the Bergman metric. We construct a fundamental domain for the action of $\mathrm{PU}(2,1;\mathbb{Z}[\omega ])$ on $H_{\mathbb{C}}^2$, 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 $\mathrm{PU}(2,1)$