A note on trace fields of complex hyperbolic groups

We show that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $Γ$ contains a loxodromic element $A$. If $A$ has eigenvalues $λ_1 = λe^{iφ},$ $λ_2 = e^{-2iφ}$, $λ_3 = λ^{-1}e^{iφ}$, we prove that $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SU}(2,1,\mathbb{Q}(Γ,λ)),$ where $\mathbb{Q}(Γ, λ)$ is the field generated by the trace field $\mathbb{Q}(Γ)$ of $Γ$ and $λ$. It follows from this that if $Γ$ is an irreducible subgroup of ${\rm SU}(2,1)$ such that the trace field $\mathbb{Q}(Γ)$ is real, then $Γ$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SO}(2,1)$. As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of ${\rm PU}(2,1)$, then $G$ is an $\mathbb{R}$-Fuchsian subgroup of ${\rm PU}(2,1)$ if and only if the invariant trace field $k(G)$ of $G$ is real.

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