Real reflections, commutators, and cross-ratios in complex hyperbolic space (Q2357413)

Summary: We provide a concrete criterion to determine whether or not two given elements of \(\mathrm{PU}(2,1)\) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups \(\mathrm{PU}(2,1,\mathcal{O}_d)\) with \(d=1,2,3,7,11\) are generated by real reflections up to index \(1, 2, 4\) or \(8\).