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

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 structure of some semi-direct product $\Gamma = \Gamma' \rtimes B$ where $\Gamma'$ is a CAT(0) group with finite-center and $B$ is a torsion-free Bieberbach group. Also, we introduce 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 $T \times \R$.