Geometric characterization of flat groups of automorphisms

作者：Baumgartner Udo^*; Schlichting Guenter; Willis George

来源：Groups, Geometry, and Dynamics, 2010, 4(1): 1-13.

DOI：10.4172/GGD/72

• Totally disconnected locally compact group
• automorphism group
• tidy subgroup
• rank
• quasi-isometry
• flat

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.

• 出版日期2010