The limit set for discrete complex hyperbolic groups

TL;DR

This paper investigates the action of discrete complex hyperbolic groups on the entire projective space, establishing a connection between equicontinuity and Kulkarni's discontinuity set, and characterizing the largest proper discontinuity domain.

Contribution

It extends the understanding of complex hyperbolic groups' actions from the unit ball to the full projective space, identifying the maximal domain of proper discontinuity.

Findings

Equicontinuity set coincides with Kulkarni's discontinuity set.

In non-elementary cases, this set is the largest domain of proper discontinuity.

Describes the discontinuity set as the complement of tangent hyperplanes at the Chen-Greenberg limit set.

Abstract

Given a discrete subgroup $\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\Bbb{H}^n_{\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $\Gamma$ on all of $ \Bbb{P}^n_{\Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ \Bbb{P}^n_{\Bbb{C}}$ which are tangent to $\partial \Bbb{H}^n_{\Bbb{C}}$ at points in the Chen-Greenberg limit set $\Lambda_{CG}(\Gamma)$.