A characterization of Fuchsian groups acting on complex hyperbolic spaces

Let G aS, SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is a",-Fuchsian; if G preserves a Lagrangian plane, then G is a"e-Fuchsian; G is Fuchsian if G is either a",-Fuchsian or a"e-Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that G is conjugate to a subgroup of S(U(1)xU(1, 1)) or SO(2, 1) if each loxodromic element in G is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a a",-Fuchsian group.