A note on trace fields of complex hyperbolic groups

TL;DR

This paper investigates the structure of irreducible subgroups of SU(2,1), showing they contain loxodromic elements and are conjugate to subgroups over specific number fields, with implications for geometric subgroup classifications.

Contribution

It establishes that irreducible subgroups of SU(2,1) contain loxodromic elements and are conjugate to subgroups over fields generated by trace and eigenvalues, linking trace field properties to geometric subgroup types.

Findings

Irreducible subgroups of SU(2,1) contain loxodromic elements.

Subgroups are conjugate to groups over fields generated by trace and eigenvalues.

Real trace fields imply conjugacy to SO(2,1) subgroups.

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.