# When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $g$ bounds a flat half plane.

$\textbf{Question:}$ Let $G$ be a torsion free CAT(0) group which is not virtually cyclic. Then if $G$ does not have a subgroup isomorphic to $\mathbb{Z}^2$, does $G$ always contain a rank one isometry?

Some remarks:

(1) The question is implied by the flat closing conjecture: $G$ contains $\mathbb{Z}^d$ iff $X$ contains a $d$-flat.

(2) When $X$ is a CAT(0) cube complex, this can be deduced by the rank rigidity theorem for CAT(0) cube groups of Caprace and Sageev in

Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal. 21 (2011), no. 4, 851–891.

(3) A weaker question one can ask is that does such $G$ always contain a free subgroup?

(4) One can of course also ask the question for any CAT(0) group or more generally groups acting on CAT(0) spaces via semi-simple isometry.

• This seems like it would follow from the rank rigidity conjecture of Ballmann and Buyalo and this result: ams.org/mathscinet-getitem?mr=3095707 – Ian Agol May 2 '16 at 14:36
• @IanAgol, thanks for the comment. But I think the rank rigidity conjecture only deals with geodesic complete CAT(0) space. I do not know in general whether a torsion free CAT(0) group always acts on a geodesic complete CAT(0) space. Maybe I missed something here. – Xiaolei Wu May 2 '16 at 15:36
• Ah, good point, I missed that hypothesis. In any case, it seems like your question is as difficult as the flat closing conjecture. – Ian Agol May 2 '16 at 16:00