A note on trace fields of complex hyperbolic groups

Heleno Cunha¹ ; Nikolay Gusevskii¹

¹Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

Groups, geometry, and dynamics, Tome 8 (2014) no. 2, pp. 355-374

Résumé

We show that if Γ is an irreducible subgroup of SU(2,1), then Γ contains a loxodromic element A. If A has eigenvalues λ1​=λeiφ, λ2​=e−2iφ, λ3​=λ−1eiφ, 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.

Détail

DOI : 10.4171/ggd/229

Classification : 32-XX
Mots-clés : Complex hyperbolic groups, trace fields