| Title:
|
A characterization of Fuchsian groups acting on complex hyperbolic spaces (English) |
| Author:
|
Fu, Xi |
| Author:
|
Li, Liulan |
| Author:
|
Wang, Xiantao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
517-525 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
$\mathbb {R}$-Fuchsian group |
| Keyword:
|
$\mathbb {C}$-Fuchsian group |
| Keyword:
|
complex line |
| Keyword:
|
$\mathbb {R}$-plane |
| Keyword:
|
trace |
| MSC:
|
20H10 |
| MSC:
|
30F35 |
| MSC:
|
30F40 |
| idZBL:
|
Zbl 1265.30182 |
| idMR:
|
MR2990191 |
| DOI:
|
10.1007/s10587-012-0026-5 |
| . |
| Date available:
|
2012-06-08T09:50:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142843 |
| . |
| Reference:
|
[1] Beardon, A. F.: The Geometry of Discrete Groups.Graduate Texts in Mathematics, Vol. 91, Springer, New York (1983). Zbl 0528.30001, MR 0698777, 10.1007/978-1-4612-1146-4 |
| Reference:
|
[2] Chen, S. S., Greenberg, L.: Hyperbolic spaces.Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers (1974), 49-87. Zbl 0295.53023, MR 0377765 |
| Reference:
|
[3] Goldman, W. M.: Complex Hyperbolic Geometry.Oxford: Clarendon Press (1999). Zbl 0939.32024, MR 1695450 |
| Reference:
|
[4] Kamiya, S.: Notes on elements of $U(1,n;\mathbb{C})$.Hiroshima Math. J. 21 (1991), 23-45. MR 1091431, 10.32917/hmj/1206128922 |
| Reference:
|
[5] Maskit, B.: Kleinian Groups.Springer-Verlag, Berlin (1988). Zbl 0627.30039, MR 0959135 |
| Reference:
|
[6] Parker, J. R., Platis, I. D.: Complex hyperbolic Fenchel-Nielsen coordinates.Topology 47 (2008), 101-135. Zbl 1169.30019, MR 2415771, 10.1016/j.top.2007.08.001 |
| Reference:
|
[7] Parker, J. R.: Notes on Complex Hyperbolic Geometry.Cambridge University Press, Preprint (2004). MR 1695450 |
| . |
