The geometry of the Eisenstein-Picard modular group

Source abstract

The Eisenstein-Picard modular group PU(2,1;Z[ω]) is defined to be the subgroup of PU(2,1) whose entries lie in the ring Z[ω], where ω is a cube root of unity. This group acts isometrically and properly discontinuously on HC2, that is, on the unit ball in C2 with the Bergman metric. We construct a fundamental domain for the action of PU(2,1;Z[ω]) on HC2, 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)