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