# Groups, Geometry, and Dynamics

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**Volume 4, Issue 1, 2010, pp. 1–13**

**DOI: 10.4171/GGD/72**

Published online: 2009-12-23

Geometric characterization of flat groups of automorphisms

Udo Baumgartner^{[1]}, Günter Schlichting

^{[2]}and George A. Willis

^{[3]}(1) University of Wollongong, Australia

(2) TU München, Garching, Germany

(3) The University of Newcastle, Callaghan, Australia

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.

*Keywords: *Totally disconnected locally compact group, automorphism group, tidy subgroup, rank, quasi-isometry, flat

Baumgartner Udo, Schlichting Günter, Willis George: Geometric characterization of flat groups of automorphisms. *Groups Geom. Dyn.* 4 (2010), 1-13. doi: 10.4171/GGD/72