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.
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Udo Baumgartner, Günter Schlichting, George A. Willis, Geometric characterization of flat groups of automorphisms. Groups Geom. Dyn. 4 (2010), no. 1, pp. 1–13DOI 10.4171/GGD/72