Geodesic
A characterization of Fuchsian groups acting on complex hyperbolic spaces
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 517-525

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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.
DOI : 10.1007/s10587-012-0026-5
Classification : 20H10, 30F35, 30F40
Keywords: $\mathbb {R}$-Fuchsian group; $\mathbb {C}$-Fuchsian group; complex line; $\mathbb {R}$-plane; trace 
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Fu, Xi; Li, Liulan; Wang, Xiantao. A characterization of Fuchsian groups acting on complex hyperbolic spaces. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 517-525. doi: 10.1007/s10587-012-0026-5

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