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