Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
$\mathbb {R}$-Fuchsian group; $\mathbb {C}$-Fuchsian group; complex line; $\mathbb {R}$-plane; trace
$\mathbb {R}$-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:
[1] Beardon, A. F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, Vol. 91, Springer, New York (1983). DOI 10.1007/978-1-4612-1146-4 | MR 0698777 | Zbl 0528.30001
[2] Chen, S. S., Greenberg, L.: Hyperbolic spaces. Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers (1974), 49-87. MR 0377765 | Zbl 0295.53023
[3] Goldman, W. M.: Complex Hyperbolic Geometry. Oxford: Clarendon Press (1999). MR 1695450 | Zbl 0939.32024
[4] Kamiya, S.: Notes on elements of $U(1,n;\mathbb{C})$. Hiroshima Math. J. 21 (1991), 23-45. DOI 10.32917/hmj/1206128922 | MR 1091431
[5] Maskit, B.: Kleinian Groups. Springer-Verlag, Berlin (1988). MR 0959135 | Zbl 0627.30039
[6] Parker, J. R., Platis, I. D.: Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47 (2008), 101-135. DOI 10.1016/j.top.2007.08.001 | MR 2415771 | Zbl 1169.30019
[7] Parker, J. R.: Notes on Complex Hyperbolic Geometry. Cambridge University Press, Preprint (2004). MR 1695450
Search
Partner of
