A characterization of Fuchsian groups acting on complex hyperbolic spaces

Title:

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Creator:

Fu, Xi, Li, Liulan, and Wang, Xiantao

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:29552b36-4118-45c4-ad9d-2f0dca4f5fd5
uuid:29552b36-4118-45c4-ad9d-2f0dca4f5fd5

Subject:

$\mathbb {R}$-Fuchsian group, $\mathbb {C}$-Fuchsian group, complex line, $\mathbb {R}$-plane, and trace

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

517-525

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:2

Harvested from:

CDK

Metadata only:

false