# Geometric characterization of flat groups of automorphisms

### Udo Baumgartner

University of Wollongong, Australia### Günter Schlichting

TU München, Garching, Germany### George A. Willis

The University of Newcastle, Callaghan, Australia

## Abstract

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.

## Cite this article

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

DOI 10.4171/GGD/72