On the semi-direct product structure of CAT(0) groups

Abstract

In this paper, we investigate finitely generated groups of isometries of CAT(0) spaces containing some central hyperbolic isometry, and study CAT(0) groups. We show that every CAT(0) group $\Gamma$ has the semi-direct product structure $\Gamma=(\cdots(((\Gamma'\rtimes\langle\delta_{n}\rangle)\rtimes\langle\delta_{n-1}\rangle)\rtimes\langle\delta_{n-2}\rangle)\cdots)\rtimes\langle\delta_{1}\rangle$ where $\Gamma'$ is a CAT(0) group with finite center and $\delta_i\in \Gamma$ for $i=1,\dots,n$, and $\Gamma$ contains a finite-index subgroup $\Gamma'\times A$ where $A$ is isomorphic to ${\mathbb{Z}}^n$. We introduce some examples and remarks. Also we provide an example of a virtually irreducible CAT(0) group with trivial-center that acts geometrically on some CAT(0) space that splits as a product $T \times {\mathbb{R}}$.