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 $\lambda_1 = \lambda e^{i\varphi}$, $\lambda_2 = e^{-2i\varphi}$, $\lambda_3 = \lambda^{-1}e^{i\varphi}$, 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.