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?
Best Answer
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.