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

TL;DR

This paper explores the algebraic structure of CAT(0) groups, revealing a semi-direct product decomposition involving groups with finite centers and free abelian factors, and provides examples illustrating these properties.

Contribution

It establishes a semi-direct product decomposition for CAT(0) groups and introduces examples of groups with specific geometric and algebraic properties.

Findings

Every CAT(0) group has a semi-direct product structure involving a group with finite center and a free abelian subgroup.

Constructs examples of CAT(0) groups with trivial center acting on spaces splitting as a product.

Shows the existence of virtually irreducible CAT(0) groups with specific geometric actions.

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 $T \times {\mathbb{R}}$.