Baumgartner, Udo and Schlichting, Günter and Willis, George (2010) *Geometric characterization of flat groups of automorphisms.* Groups .

## Abstract

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 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 ton-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.(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 ton-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.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | totally disconnected locally compact group, automorphism group, tidy subgroup, rank, quasi-isometry, flat |

Subjects: | UNSPECIFIED |

Faculty: | UNSPECIFIED |

Depositing User: | Dr David Allingham |

Date Deposited: | 28 Sep 2012 12:05 |

Last Modified: | 28 Sep 2012 12:53 |

URI: | https://docserver.carma.newcastle.edu.au/id/eprint/1181 |

### Actions (login required)

View Item |