Geometric characterization of flat groups of automorphisms

TL;DR

This paper explores the geometric properties of flat automorphism groups of totally disconnected, locally compact groups, establishing a correspondence between orbit quasi-isometry to Euclidean space and flatness of automorphism subgroups.

Contribution

It proves a partial converse relating orbit quasi-isometry to flat automorphism subgroups and identifies a natural flat subgroup within these automorphism groups.

Findings

Orbits of flat automorphism groups are quasi-isometric to Euclidean space.

Groups with orbits quasi-isometric to Euclidean space contain a finite index flat subgroup.

The paper extends the understanding of the geometric structure of automorphism 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 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.