Do all infinite CAT(0) groups contain a $\mathbb{Z}$ subgroup? I am aware that this has been established for hyperbolic groups, and similar questions have appeared on open questions lists for CAT(0) groups, namely here. However, typically this is worded as 'Does there exist an infinite torsion subgroup of a CAT(0) group' or 'is there a infinite torsion group which acts properly discontinuously on a CAT(0) space.' Obviously subgroups can be fairly distorted and cocompactness is a strong condition. Does anyone know if this is an established fact, or is this still an open question?
For the sake of clarity I am defining $G$ to be CAT(0) if it acts properly discontinuously and cocompactly on a proper CAT(0) space.
It is proven by Swenson in
E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358.
that each infinite CAT(0) group contains an infinite order element.