The geometry of the Eisenstein-Picard modular group

Abstract

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

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