Let G ⊂ SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ℂ-Fuchsian; if G preserves a Lagrangian plane, then G is ℝ-Fuchsian; G is Fuchsian if G is either ℂ-Fuchsian or ℝ-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 tha...
