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

Abstract

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Let G 𝐒𝐔 ( 2 , 1 ) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is -Fuchsian; if G preserves a Lagrangian plane, then G is -Fuchsian; G is Fuchsian if G is either -Fuchsian or -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 𝐒 ( U ( 1 ) × U ( 1 , 1 ) ) or 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 -Fuchsian group.

How to cite

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Fu, 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

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  4. Kamiya, S., Notes on elements of U ( 1 , n ; ) , Hiroshima Math. J. 21 (1991), 23-45. (1991) MR1091431
  5. Maskit, B., Kleinian Groups, Springer-Verlag, Berlin (1988). (1988) Zbl0627.30039MR0959135
  6. 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
  7. Parker, J. R., Notes on Complex Hyperbolic Geometry, Cambridge University Press, Preprint (2004). (2004) MR1695450

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