Ballmann–Buyalo periodic rank-one geodesics conjecture

Establish that for every locally compact CAT(0) space X with a geometric action by a group G, if X contains a rank-1 geodesic (i.e., a complete geodesic that does not bound a flat half-plane), then X contains a G-periodic rank-1 geodesic.

Background

Section 5.1 discusses classifying CAT(0) spaces by rank and introduces an open conjecture of Ballmann and Buyalo concerning the existence of periodic rank-1 geodesics. A geodesic is rank-1 if it does not bound a flat half-plane, and a CAT(0) space is rank-one if it contains such a geodesic.

The paper later leverages rank rigidity results (e.g., Caprace–Sageev’s theorem for finite-dimensional CAT(0) cube complexes) to obtain Morse elements under certain boundary conditions, highlighting the importance of periodic rank-1 geodesics in understanding group actions and boundary dynamics.