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.