Picard modular groups generated by complex reflections

Abstract

In this short note we use the presentations found by the various authors to show that the Picard modular groupsPU⁡(2,1,Od)\operatorname {PU}(2,1,\mathcal {O}_d)withd=1,3,7d=1,3,7(respectively the quaternion hyperbolic latticePSp⁡(2,1,H)\operatorname {PSp}(2,1,\mathcal {H})with entries in the Hurwitz integer ringH\mathcal {H}) are generated by complex (resp. quaternionic) reflections, and that the Picard modular groupsPU⁡(2,1,Od)\operatorname {PU}(2,1,\mathcal {O}_d)withd=2,11d=2,11have an index 4 subgroup generated by complex reflections.