# Article

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Keywords:
$\mathbb {R}$-Fuchsian group; $\mathbb {C}$-Fuchsian group; complex line; $\mathbb {R}$-plane; trace
Summary:
Let $G\subset {\bf SU}(2,1)$ be a non-elementary complex hyperbolic Kleinian group. If $G$ preserves a complex line, then $G$ is $\mathbb {C}$-Fuchsian; if $G$ preserves a Lagrangian plane, then $G$ is $\mathbb {R}$-Fuchsian; $G$ is Fuchsian if $G$ is either $\mathbb {C}$-Fuchsian or $\mathbb {R}$-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 that $G$ is conjugate to a subgroup of ${\bf S}(U(1)\times U(1,1))$ or ${\bf SO}(2,1)$ if each loxodromic element in $G$ is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a $\mathbb {C}$-Fuchsian group.
References:
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