Two dimensional Veronese groups with an invariant ball

Abstract

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in ${\mathbb{P}}_{\mathbb{C}}^2$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation in $\PSL(3,\Bbb{C})$ with a non-empty region of discontinuity which is not conjugate to a complex hyperbolic subgroup. This provides a way to construct new examples of Kleinian groups acting on ${\mathbb{P}}_{\mathbb{C}}^2$.