Elements of infinite order in CAT(0) groups

Question

In

E. Swenson, A cut point theorem for CAT(0) groups, J. Differential Geom. 53 (1999), no. 2, 327–358.

the author shows (Theorem 11) that if a group $G$ acts geometrically (i.e. properly discontinuously) and cocompactly by isometries on a CAT(0) space, then $G$ has an element of infinite order. However, every finite group is a CAT(0) group.

So I'm confused. Could specialist comment on this?

Answer 1

The author is implicitly assuming that the CAT(0) space is unbounded in the first sentence of the proof of theorem 11 when they say "choose a geodesic ray". In the first paragraph of the section they say that rays are parametrized on $[0,\infty)$. In particular $G$ can not act geometrically on the space if it is finite.

Of course, as you point out, finite groups are CAT(0) in a trivial way, in that they act on (bounded) compact CAT(0) spaces geometrically. Although, this is a completely trivial uninteresting case so from the geometric perspective, so people don't really think about it.

Answer 2

It seems the proof is already assuming the group is infinite. Certainly it is true for infinite groups (this is well-known). Maybe somewhere the author mentions he's only talking about infinite groups?