A note on trace fields of complex hyperbolic groups

作者：Cunha Heleno^*; Gusevskii Nikolay

来源：Groups, Geometry, and Dynamics, 2014, 8(2): 355-374.

DOI：10.4171/GGD/229

• Complex hyperbolic groups
• trace fields

We show that if Gamma is an irreducible subgroup of 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 SU(2,1) to a subgroup of SU(2,1, Q(Gamma,lambda)), where Q (Gamma,lambda) is the field generated by the trace field Q(Gamma) of Gamma and lambda. It follows from this that if Gamma is an irreducible subgroup of SU(2,1) such that the trace field Q(Gamma) is real, then 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 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.