The geometry of the Eisenstein-Picard modular group

The Eisenstein-Picard modular group PU(2, 1; Z[omega]) is defined to be the subgroup of PU(2, 1) whose entries lie in the ring Z[omega], where omega is a cube root of unity. This group acts isometrically and properly discontinuously on H-C(2), that is, on the unit ball in C-2 with the Bergman metric. We construct a fundamental domain for the action of PU(2, 1; Z[omega]) on H-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 PU(2, 1).