Local link criteria preventing cyclic splittings

Develop a condition, stated purely in terms of links of vertices in a compact non-positively curved cube complex X with π1(X) one-ended, that guarantees π1(X) does not split over Z.

Background

The paper establishes combinatorial criteria (via Whitehead complexes and bounds on k-cuts) in the hyperbolic setting to preclude cyclic splittings, but lacks a local link-based criterion akin to Whitehead’s lemma for free splittings.

A local combinatorial criterion would extend the algorithmic and geometric tools to detect absence of cyclic splittings in broader settings, potentially beyond hyperbolic cube complexes.