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

Julien Paupert (1) , Pierre Will (2)

1 School of Mathematical and Statistical Sciences (Arizona, Tempe)
2 IF - Institut Fourier

We provide a concrete criterion to determine whether or not two given elements of 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 ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8.

Mots clés: Complex hyperbolic geometry; groups generated by reflection; groups generated by reflection.; isometries of a symmetric space

Domaines: Topologie géométrique [math.GT]

Julien Paupert, Pierre Will. Real reflections, commutators and cross-ratios in complex hyperbolic space. Groups, Geometry, and Dynamics, 2017, 11 (1), pp.311-352. ⟨hal-00916997⟩