Geometric characterization of flat groups of automorphisms

Abstract

If \mathcal H is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G , then each orbit of \mathcal H in the metric space \mathcal 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 \mathcal B(G) is a proper metric space and let \mathcal H be a group of automorphisms of G such that some (equivalently every) orbit of \mathcal H in \mathcal B(G) is quasi-isometric to n -dimensional Euclidean space, then \mathcal 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.