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

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

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