Picard modular groups generated by complex reflections

Abstract

In this short note we use the presentations found in \cite{MP} and \cite{Po} to show that the Picard modular groups $\mathrm{PU}(2,1,{\mathcal{O}}_d)$ with $d=1,3,7$ (respectively the quaternion hyperbolic lattice $\mathrm{PSp}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$) are generated by complex (resp. quaternionic) reflections, and that the Picard modular groups $\mathrm{PU}(2,1,{\mathcal{O}}_d)$ with $d=2,11$ have an index 4 subgroup generated by complex reflections.

5 pages