A note on trace fields of complex hyperbolic groups

Abstract

We show that if $\Gamma$ is an irreducible subgroup of SU(2,1), then $\Gamma$ contains a loxodromic element $A$. If $A$ has eigenvalues ${\lambda }_1=\lambda e^{i\phi }$, ${\lambda }_2=e^{-2i\phi }$, ${\lambda }_3={\lambda }^{-1}{e}^{i\phi }$, we prove that $\Gamma$ is conjugate in SU(2,1) to a subgroup of 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 SU(2,1) such that the trace field $\mathbb{Q}(\Gamma )$ is real, then $\Gamma$ is conjugate in SU(2,1) to a subgroup of SO(2,1). As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of PU(2,1), then $G$ is an $\mathbb{R}$-Fuchsian subgroup of PU(2,1) if and only if the invariant trace field $k(G)$ of $G$ is real.