Ballmann–Buyalo periodic rank-one geodesic conjecture

Let $X$ be a locally compact $\operatorname{CAT}(0)$ space, and let $G$ act geometrically on $X$. A complete geodesic has rank 1 if it does not bound a flat half-plane.

Ballmann–Buyalo's conjecture. If $X$ contains a geodesic of rank $1$, then it also contains a $G$-periodic geodesic of rank $1$.

This conjecture asserts that a rank-one geodesic in a locally compact $\operatorname{CAT}(0)$ space with a geometric group action can be chosen periodic. It is presented as an open conjecture due to Ballmann and Buyalo; the paper studies related existence questions for finite-dimensional proper $\operatorname{CAT}(0)$ cube complexes.