## Osaka Journal of Mathematics

### Decomposition of complex hyperbolic isometries by two complex symmetries

#### Abstract

Let $\mathbf{PU}(2,1)$ denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space $\mathbf{H}_{\mathbb{C}}^{2}$, and the group $\mathbf{SU}(2,1)$ is a 3-fold covering of $\mathbf{PU}(2,1)$: $\mathbf{PU}(2,1)=\mathbf{SU}(2,1)/\{\omega I:\omega^{3}=1\}$. We study how to decompose a given pair of isometries $(A,B)\in \mathbf{SU}(2,1)^{2}$ under the form $A=I_{1}I_{2}$ and $B=I_{3}I_{2},$ where the $I_{k}$'s are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is $\mathbb{C}$-decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].

#### Article information

Source
Osaka J. Math. Volume 54, Number 4 (2017), 661-677.

Dates
First available in Project Euclid: 20 October 2017

https://projecteuclid.org/euclid.ojm/1508486570

Zentralblatt MATH identifier
06821130

Subjects
Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations
Secondary: 20E45: Conjugacy classes

#### Citation

Ren, Xue-Jing; Xie, Bao-Hua; Jiang, Yue-Ping. Decomposition of complex hyperbolic isometries by two complex symmetries. Osaka J. Math. 54 (2017), no. 4, 661--677.https://projecteuclid.org/euclid.ojm/1508486570

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