The Limit Set for Discrete Complex Hyperbolic Groups

We suppose a discrete subgroup Γ of PU(1,n) that acts by isometries on the unit complex ball $\mathbb{H}^n_\mathbb{C}$. In this setting, much work has been done in order to understand the action of the group. However, when we look at the action of Γ on all of $\mathbb{P}^n_\mathbb{C}$, little or nothing is known. In this paper, we study the action in the entire projective space, and are able to show that its equicontinuity agrees with its Kulkarni discontinuity set. Moreover, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously. It can be described as the complement of the union of all complex projective hyperplanes in $\mathbb{P}^n_\mathbb{C}$ that are tangent to $\partial \mathbb{H}^n_\mathbb{C}$ at points in the Chen-Greenberg limit set ΛCG(Γ).