Regular elements in CAT(0) groups

TL;DR

This paper proves that groups acting properly and cocompactly on CAT(0) spaces contain elements acting hyperbolically on each indecomposable factor, implying the existence of free abelian subgroups related to the space's product structure.

Contribution

It establishes the presence of hyperbolic elements in groups acting on CAT(0) spaces and links the group's algebraic structure to the geometric decomposition of the space.

Findings

G contains hyperbolic elements on each indecomposable factor

If the space is a product of d factors, G contains a free abelian group of rank d

The result connects geometric decomposition with algebraic subgroup structure

Abstract

Let X be a locally compact geodesically complete CAT(0) space and G be a discrete group acting properly and cocompactly on X. We show that G contains an element acting as a hyperbolic isometry on each indecomposable de Rham factor of X. It follows that if X is a product of d factors, then G contains free abelian group of rank d.