Key points are not available for this paper at this time.
Halin's well-known grid theorem states that a graph G with a thick end must contain a subdivision of the hexagonal half-grid. We obtain the following strengthening when G is vertex-transitive and locally finite. Either G is quasi-isometric to a tree (and therefore has no thick end), or it contains a subdivision of the full hexagonal grid.
Georgakopoulos et al. (Sat,) studied this question.