Building prescribed quantitative orbit equivalence with the integers

Abstract

Two groups are orbit equivalent if they both admit an action on a same probability space that share the same orbits. In particular, the Ornstein–Weiss theorem implies that all infinite countable amenable groups are orbit equivalent to the group of integers. To refine this notion between infinite countable amenable groups, Delabie, Koivisto, Le Maître and Tessera introduced a quantitative version of orbit equivalence. They furthermore obtained obstructions to the existence of such equivalence using the isoperimetric profile. In this article, we offer to answer the inverse problem (find a group being orbit equivalent to a prescribed group with prescribed quantification) in the case of the group of integers using the so called Følner tiling shifts introduced by Delabie et al. To do so, we use the diagonal products defined by Brieussel and Zheng giving groups with prescribed isoperimetric profile.