On the Equicontinuity Region of Discrete Subgroups of PU(1,n)

Abstract

Let $G$ be a discrete subgroup of PU(1,n). Then $G$ acts on ${\mathbb{P}}_{\mathbb{C}}^n$ preserving the unit ball ${\mathbb{H}}_{\mathbb{C}}^n$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in ${\mathbb{P}}_{\mathbb{C}}^n$: It is the complement of the union of all complex projective hyperplanes in ${\mathbb{P}}_{\mathbb{C}}^n$ which are tangent to $\partial {\mathbb{H}}_{\mathbb{C}}^n$ at points in the Chen-Greenberg limit set ${\Lambda }_{CG}(G)$, a closed $G$-invariant subset of $\partial {\mathbb{H}}_{\mathbb{C}}^n$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.