Regularity of limit sets of AdS quasi-Fuchsian groups

TL;DR

This paper investigates the geometric regularity of limit sets of AdS quasi-Fuchsian groups, revealing they are Lipschitz but not differentiable unless Fuchsian, and shows non-Fuchsian groups are Zariski dense.

Contribution

It proves that limit sets of AdS quasi-Fuchsian groups are never  smooth except for Fuchsian groups and establishes Zariski density of non-Fuchsian groups in PO(n,2).

Findings

Limit sets are Lipschitz submanifolds.

Limit sets are not  smooth unless Fuchsian.

Non-Fuchsian groups are Zariski dense in PO(n,2).

Abstract

Limit sets of $\mathrm{AdS}$-quasi-Fuchsian groups of $\mathrm{PO}(n,2)$ are always Lipschitz submanifolds. The aim of this article is to show that they are never $\mathcal{C}^1$, except for the case of Fuchsian groups. As a byproduct we show that $\mathrm{AdS}$-quasi-Fuchsian groups that are not Fuchsian are Zariski dense in $\mathrm{PO}(n,2)$.