A note on trace fields of complex hyperbolic groups

Abstract

We show that if $\Gamma$ is an irreducible subgroup of ${\rm SU}(2,1)$, then $\Gamma$ contains a loxodromic element $A$. If $A$ has eigenvalues $\lambda_1 = \lambda e^{i\varphi},$ $\lambda_2 = e^{-2i\varphi}$, $\lambda_3 = \lambda^{-1}e^{i\varphi}$, we prove that $\Gamma$ is conjugate in ${\rm SU}(2,1)$ to a subgroup of ${\rm SU}(2,1,\mathbb{Q}(\Gamma,\lambda)),$ where $\mathbb{Q}(\Gamma, \lambda)$ is the field generated by the trace field $\mathbb{Q}(\Gamma)$ of $\Gamma$ and $\lambda$. It follows from this that if $\Gamma$ is an irreducible subgroup of ${\rm SU}(2,1)$ such that the trace field $\mathbb{Q}(\Gamma)$ is real, then $\Gamma$ 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.