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.
Udo Baumgartner
1
;
Günter Schlichting
2
;
George A. Willis
3
1
University of Wollongong, Australia
2
TU München, Garching, Germany
3
The University of Newcastle, Callaghan, Australia
Udo Baumgartner; Günter Schlichting; George A. Willis. Geometric characterization of flat groups of automorphisms. Groups, geometry, and dynamics, Tome 4 (2010) no. 1, pp. 1-13. doi: 10.4171/ggd/72
@article{10_4171_ggd_72,
author = {Udo Baumgartner and G\"unter Schlichting and George A. Willis},
title = {Geometric characterization of flat groups of automorphisms},
journal = {Groups, geometry, and dynamics},
pages = {1--13},
year = {2010},
volume = {4},
number = {1},
doi = {10.4171/ggd/72},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/72/}
}
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AU - Günter Schlichting
AU - George A. Willis
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