A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs

We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu. As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu.