Isometric orbit equivalence for probability-measure preserving actions

TL;DR

This paper introduces and explores isometric orbit equivalence for p.m.p. actions of finitely generated groups, revealing rigidity in some cases and constructing nontrivial examples, including for free groups.

Contribution

It defines isometric orbit equivalence for p.m.p. actions, proves rigidity results for certain groups, and constructs new nontrivial examples demonstrating its properties.

Findings

Isometric orbit equivalence is rigid for groups with Cayley graphs having a countable automorphism group.

Constructs nontrivial examples of isometric orbit equivalent actions for free groups.

Shows that mixing property is not preserved under isometric orbit equivalence.

Abstract

We study probability-measure preserving (p.m.p.) actions of finitely generated groups via the graphings they define. We introduce and study the notion of isometric orbit equivalence for p.m.p. actions: two p.m.p. actions are isometric orbit equivalent if the graphings defined by some fixed generating systems of the groups are measurably isometric. We highlight two kind of phenomena. First, we prove that the notion of isometric orbit equivalence is rigid for groups whose Cayley graph, with respect to a fixed generating system, has a countable group of automorphism. On the other hand, we introduce a general construction of isometric orbit equivalent p.m.p. actions, which leads to interesting nontrivial examples of isometric orbit equivalent p.m.p. actions for the free group. In particular, our examples show that mixing is not invariant under isometric orbit equivalence.