  If H is a flat group of automorphisms of finite rank n of a totally
disconnected, locally compact group G, then each orbit of H in the metric space
B(G) of compact, open subgroups of G is quasi-isometric to n-dimensional
euclidean space. In this note we prove the following partial converse: Assume
that G is a totally disconnected, locally compact group such that B(G) is a
proper metric space and let H be a group of automorphisms of G such that some
(equivalently every) orbit of H in B(G) is quasi-isometric to n-dimensional
euclidean space, then H has a finite index subgroup which is flat of rank n. We
can draw this conclusion under weaker assumptions. We also single out a
naturally defined flat subgroup of such groups of automorphisms.
