A characterization of Fuchsian groups acting on complex hyperbolic spaces

Fu, Xi ; Li, Liulan ; Wang, Xiantao

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 517-525.

Résumé

Let $G\subset \mathbf{S}\mathbf{U}(2,1)$ 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 $\mathbf{S}(U(1)\times U(1,1))$ or $\mathbf{S}\mathbf{O}(2,1)$ 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.

Détail

DOI : 10.1007/s10587-012-0026-5

Classification : 20H10, 30F35, 30F40
Mots-clés : $\mathbb{R}$-Fuchsian group; $\mathbb{C}$-Fuchsian group; complex line; $\mathbb{R}$-plane; trace