JournalsggdVol. 8, No. 2pp. 355–374

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
A note on trace fields of complex hyperbolic groups cover
Download PDF

Abstract

We show that if Γ\Gamma is an irreducible subgroup of SU(2,1), then Γ\Gamma contains a loxodromic element AA. If AA has eigenvalues λ1=λeiφ\lambda_1 = \lambda e^{i\varphi}, λ2=e2iφ\lambda_2 = e^{-2i\varphi}, λ3=λ1eiφ\lambda_3 = \lambda^{-1}e^{i\varphi}, we prove that Γ\Gamma is conjugate in SU(2,1) to a subgroup of SU(2,1,Q(Γ,λ))(2,1,\mathbb{Q}(\Gamma,\lambda)), where Q(Γ,λ)\mathbb{Q}(\Gamma, \lambda) is the field generated by the trace field Q(Γ)\mathbb{Q}(\Gamma) of Γ\Gamma and λ\lambda. It follows from this that if Γ\Gamma is an irreducible subgroup of SU(2,1) such that the trace field Q(Γ)\mathbb{Q}(\Gamma) is real, then Γ\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 GG is an irreducible discrete subgroup of PU(2,1), then GG is an R\mathbb R-Fuchsian subgroup of PU(2,1) if and only if the invariant trace field k(G)k(G) of GG 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