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 is a flat group of automorphisms of finite rank of a totally disconnected, locally compact group , then each orbit of in the metric space of compact, open subgroups of is quasi-isometric to -dimensional Euclidean space. In this note we prove the following partial converse: Assume that is a totally disconnected, locally compact group such that is a proper metric space and let be a group of automorphisms of such that some (equivalently every) orbit of in is quasi-isometric to -dimensional Euclidean space, then has a finite index subgroup which is flat of rank . 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