A note on trace fields of complex hyperbolic groups

We show that if $\mathrm{\Gamma }$ is an irreducible subgroup of $\mathrm{S}\mathrm{U}(2,1)$, then $\mathrm{\Gamma }$ contains a loxodromic element $A$. If $A$ has eigenvalues $\lambda_1 = \lambda e^{i\varphi},$ ${\lambda }_2=e^{-2i\phi }$, $\lambda_3 = \lambda^{-1}e^{i\varphi}$,weprovethat$\Gamma$isconjugatein${\rm SU}(2,1)$toasubgroupof${\rm SU}(2,1,$\mathbb{Q}$(\Gamma,\lambda)),$where$$\mathbb{Q}$(\Gamma, \lambda)$isthefieldgeneratedbythetracefield$$\mathbb{Q}$(\Gamma)$of$\Gamma$and$\lambda$\mathrm{.}Itfollowsfromthisthatif$\Gamma$isanirreduciblesubgroupof${\rm SU}(2,1)$ such that the trace field $\mathbb{Q}(\mathrm{\Gamma })$ is real, then $\mathrm{\Gamma }$ is conjugate in ${\rm SU}(2,1)$toasubgroupof${\rm SO}(2,1)$. As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of ${\rm PU}(2,1)$,then$G$isan$$\mathbb{R}$$-Fuchsiansubgroupof${\rm PU}(2,1)$ if and only if the invariant trace field $k(G)$ of $G$ is real.