Geometric characterization of flat groups of automorphisms

Details Geometric characterization of flat groups of automorphisms Geometric characterization of flat groups of automorphisms

2008-07-31

arxiv.org/abs/0807.5060v1

Mathematics

Group Theory

12 pages, submitted

Scientific paper

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.

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