Two dimensional Veronese groups with an invariant ball

TL;DR

This paper characterizes complex hyperbolic groups preserving a Veronese curve in projective space and explores deformations of surface groups into complex projective groups, leading to new Kleinian group examples.

Contribution

It provides a classification of groups preserving a Veronese curve and demonstrates deformations of surface groups into complex projective groups with non-trivial discontinuity regions.

Findings

Characterization of complex hyperbolic groups with invariant Veronese curve

Existence of deformations of surface groups into PSL(3,C)

Construction of new Kleinian group examples

Abstract

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. 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 $\Bbb{P}^2_\Bbb{C}$.