A characterization of complex hyperbolic Kleinian groups in dimension 3 with trace fields contained in $\mathbb{R}$

Abstract

We show that $\Gamma <\text{SU}(3,1)$ is a non-elementary complex hyperbolic Kleinian group in which $tr(\gamma )\in \mathbb{R}$ for all $\gamma \in \Gamma$ if and only if $\Gamma$ is conjugate to a subgroup of $\text{SO}(3,1)$ or $\text{SU}(1,1)\times \text{SU}(2)$.