# A characterization of Fuchsian groups acting on complex hyperbolic spaces

Xi Fu; Liulan Li; Xiantao Wang

Czechoslovak Mathematical Journal (2012)

- Volume: 62, Issue: 2, page 517-525
- ISSN: 0011-4642

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topFu, Xi, Li, Liulan, and Wang, Xiantao. "A characterization of Fuchsian groups acting on complex hyperbolic spaces." Czechoslovak Mathematical Journal 62.2 (2012): 517-525. <http://eudml.org/doc/246750>.

@article{Fu2012,

abstract = {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.},

author = {Fu, Xi, Li, Liulan, Wang, Xiantao},

journal = {Czechoslovak Mathematical Journal},

keywords = {$\mathbb \{R\}$-Fuchsian group; $\mathbb \{C\}$-Fuchsian group; complex line; $\mathbb \{R\}$-plane; trace; -Fuchsian group; -Fuchsian group; complex line; -plane; trace},

language = {eng},

number = {2},

pages = {517-525},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A characterization of Fuchsian groups acting on complex hyperbolic spaces},

url = {http://eudml.org/doc/246750},

volume = {62},

year = {2012},

}

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

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 62

IS - 2

SP - 517

EP - 525

AB - 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.

LA - eng

KW - $\mathbb {R}$-Fuchsian group; $\mathbb {C}$-Fuchsian group; complex line; $\mathbb {R}$-plane; trace; -Fuchsian group; -Fuchsian group; complex line; -plane; trace

UR - http://eudml.org/doc/246750

ER -

## References

top- Beardon, A. F., 10.1007/978-1-4612-1146-4, Graduate Texts in Mathematics, Vol. 91, Springer, New York (1983). (1983) Zbl0528.30001MR0698777DOI10.1007/978-1-4612-1146-4
- Chen, S. S., Greenberg, L., Hyperbolic spaces, Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers (1974), 49-87. (1974) Zbl0295.53023MR0377765
- Goldman, W. M., Complex Hyperbolic Geometry, Oxford: Clarendon Press (1999). (1999) Zbl0939.32024MR1695450
- Kamiya, S., Notes on elements of $U(1,n;\u2102)$, Hiroshima Math. J. 21 (1991), 23-45. (1991) MR1091431
- Maskit, B., Kleinian Groups, Springer-Verlag, Berlin (1988). (1988) Zbl0627.30039MR0959135
- Parker, J. R., Platis, I. D., 10.1016/j.top.2007.08.001, Topology 47 (2008), 101-135. (2008) Zbl1169.30019MR2415771DOI10.1016/j.top.2007.08.001
- Parker, J. R., Notes on Complex Hyperbolic Geometry, Cambridge University Press, Preprint (2004). (2004) MR1695450

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