Geodesic
A characterization of Fuchsian groups acting on complex hyperbolic spaces
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 517-525
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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 
@article{10_1007_s10587_012_0026_5,
     author = {Fu, Xi and Li, Liulan and Wang, Xiantao},
     title = {A characterization of {Fuchsian} groups acting on complex hyperbolic spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {517--525},
     year = {2012},
     volume = {62},
     number = {2},
     doi = {10.1007/s10587-012-0026-5},
     mrnumber = {2990191},
     zbl = {1265.30182},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0026-5/}
}
TY  - JOUR
AU  - Fu, Xi
AU  - Li, Liulan
AU  - Wang, Xiantao
TI  - A characterization of Fuchsian groups acting on complex hyperbolic spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2012
SP  - 517
EP  - 525
VL  - 62
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0026-5/
DO  - 10.1007/s10587-012-0026-5
LA  - en
ID  - 10_1007_s10587_012_0026_5
ER  - 
%0 Journal Article
%A Fu, Xi
%A Li, Liulan
%A Wang, Xiantao
%T A characterization of Fuchsian groups acting on complex hyperbolic spaces
%J Czechoslovak Mathematical Journal
%D 2012
%P 517-525
%V 62
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-012-0026-5/
%R 10.1007/s10587-012-0026-5
%G en
%F 10_1007_s10587_012_0026_5
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

[1] Beardon, A. F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics, Vol. 91, Springer, New York (1983). | DOI | MR | Zbl

[2] Chen, S. S., Greenberg, L.: Hyperbolic spaces. Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers (1974), 49-87. | MR | Zbl

[3] Goldman, W. M.: Complex Hyperbolic Geometry. Oxford: Clarendon Press (1999). | MR | Zbl

[4] Kamiya, S.: Notes on elements of $U(1,n;\mathbb{C})$. Hiroshima Math. J. 21 (1991), 23-45. | DOI | MR

[5] Maskit, B.: Kleinian Groups. Springer-Verlag, Berlin (1988). | MR | Zbl

[6] Parker, J. R., Platis, I. D.: Complex hyperbolic Fenchel-Nielsen coordinates. Topology 47 (2008), 101-135. | DOI | MR | Zbl

[7] Parker, J. R.: Notes on Complex Hyperbolic Geometry. Cambridge University Press, Preprint (2004). | MR

Cité par Sources :