A characterization of Fuchsian groups acting on complex hyperbolic spaces
A characterization of Fuchsian groups acting on complex hyperbolic spaces
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DOI：
10.1007/s10587-012-0026-5
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发表时间：
2012-10
影响因子：
0.5
通讯作者：
Xi Fu;Liulan Li;Xiantao Wang
中科院分区：
文献类型：
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作者：
Xi Fu;Liulan Li;Xiantao Wang
LetG∧SU（2,1）是一个非初等复双曲克莱因群。如果保留一条复线，则英文是-Fuchsian；如果它保持拉格朗日平面，则它是f -Fuchsian；Gis -Fuchsian如果Gis要么是-Fuchsian要么是-Fuchsian。在本文中，我们证明了如果所有元素的轨迹都是实数，则英语是富克斯的。这是B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988，在复双曲等距群的情况下，定理V.G. 18的类似结果。作为我们的主要结果的一个应用，我们证明了如果每个双曲元素ings是双曲的，则gis共轭到s (U（1)×U(1,1)）或so（2,1）的子群。此外，通过给出一个-Fuchsian群，我们证明了我们的主要结果的逆成立。
LetG⊂SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. IfGpreserves a complex line, thenGis ℂ-Fuchsian; ifGpreserves a Lagrangian plane, thenGis ℝ-Fuchsian;Gis Fuchsian ifGis either ℂ-Fuchsian or ℝ-Fuchsian. In this paper, we prove that if the traces of all elements inGare real, thenGis 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 thatGis conjugate to a subgroup ofS(U(1)×U(1, 1)) orSO(2, 1) if each loxodromic element inGis hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a ℂ-Fuchsian group.