Isometric orbit equivalence for probability-measure preserving actions

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.