# A note on trace fields of complex hyperbolic groups

### Heleno Cunha

Universidade Federal de Minas Gerais, Belo Horizonte, Brazil### Nikolay Gusevskii

Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

## Abstract

We show that if $Γ$ is an irreducible subgroup of SU(2,1), then $Γ$ contains a loxodromic element $A$. If $A$ has eigenvalues $λ_{1}=λe_{iφ}$, $λ_{2}=e_{−2iφ}$, $λ_{3}=λ_{−1}e_{iφ}$, we prove that $Γ$ is conjugate in SU(2,1) to a subgroup of SU$(2,1,Q(Γ,λ))$, where $Q(Γ,λ)$ is the field generated by the trace field $Q(Γ)$ of $Γ$ and $λ$. It follows from this that if $Γ$ is an irreducible subgroup of SU(2,1) such that the trace field $Q(Γ)$ is real, then $Γ$ 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 $R$-Fuchsian subgroup of PU(2,1) if and only if the invariant trace field $k(G)$ of $G$ is real.

## Cite this article

Heleno Cunha, Nikolay Gusevskii, A note on trace fields of complex hyperbolic groups. Groups Geom. Dyn. 8 (2014), no. 2, pp. 355–374

DOI 10.4171/GGD/229