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

TL;DR

This paper characterizes the equicontinuity region of discrete subgroups of PU(1,n) acting on complex projective space, showing it as the complement of tangent hyperplanes at the Chen-Greenberg limit set, and proves the action is discontinuous there.

Contribution

It precisely determines the equicontinuity region for these groups and establishes the discontinuity of the action on this region, extending understanding of their dynamics.

Findings

Equicontinuity region is the complement of tangent hyperplanes at the limit set.

Action on the equicontinuity region is discontinuous.

Provides a geometric description of the equicontinuity region.

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.