| Résume | Résumé : Two groups G and H are orbit equivalent if there exist two free probability measure-preserving G- and H-actions on a standard probability space, having the same orbits. Since this theory is trivial among infinite amenable groups, we need to strengthen its definition.
A well-known invariant of quasi-isometry, called the isoperimetric profile, provides obstructions to quantitative strengthenings of orbit equivalence. It also serves as a measurement of amenability. The more a group is amenable, the faster its profile tends to infinity.
Quantitative orbit equivalence thus quantifies, in some sense, how much the geometries of amenable non-quasi-isometric groups differ.
Our work focuses on a class of groups which look like lamplighters: lampshuffler groups. Our main results are a computation of their isoperimetric profiles and a classification up to quantitative orbit equivalence of lampshufflers over free abelian groups |
