A note on trace fields of complex hyperbolic groups

Abstract

We show that if $\Gamma$ is an irreducible subgroup of $\mathrm{SU}(2,1)$, 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 $\mathrm{SU}(2,1)$ to a subgroup of $\mathrm{SU}(2,1,\mathbb{Q}(\Gamma ,\lambda )),$ 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 $\mathrm{SU}(2,1)$ such that the trace field $\mathbb{Q}(\Gamma )$ is real, then $\Gamma$ is conjugate in $\mathrm{SU}(2,1)$ to a subgroup of $\mathrm{SO}(2,1)$. As a geometric application of the above, we get that if $G$ is an irreducible discrete subgroup of $\mathrm{PU}(2,1)$, then $G$ is an $\mathbb{R}$-Fuchsian subgroup of $\mathrm{PU}(2,1)$ if and only if the invariant trace field $k(G)$ of $G$ is real.

To appear in Groups, Geometry, and Dynamics