Is amenability invariant under orbit equivalence?

1. What people mean by “amenability” in this discussion

“Amenability” appears in at least three related settings in the cited literature: (i) amenability of a discrete countable group (classical Følner/mean definition), (ii) amenability (aka hyperfiniteness) of a countable p.m.p. equivalence relation, and (iii) amenability of an action or groupoid (fixed-point / invariant mean formulations). The equivalence-relation notion is often phrased as hyperfiniteness: a discrete p.m.p. equivalence relation is amenable iff it is hyperfinite (i.e., an increasing union of finite-class relations mod-null sets) ^{[5] [1]}.

2. Free actions of amenable groups: orbit relations are amenable

The classical Connes–Feldman–Weiss / Ornstein–Weiss results show that any Borel action of a countable amenable group yields, after ignoring a null set, a hyperfinite (hence amenable) orbit equivalence relation; in short, amenable groups give rise to amenable orbit relations in the standard p.m.p. setting ^[1].

3. Non‑amenable groups can produce amenable orbit relations

Conversely, there are explicit constructions where a non‑amenable group acts so that the resulting orbit equivalence relation is amenable (for example via actions with large stabilizers or self-similar/tree boundary actions whose orbit relation sits inside a hyperfinite tail relation) — this shows amenability of the orbit relation does not force the acting group to be amenable ^{[2] [3]}. MathOverflow discussion and the groupoid literature highlight concrete examples and mechanisms where the group is non‑amenable but the orbit relation is hyperfinite ^{[2] [3]}.

4. Invariance under (orbit) equivalence: it depends on hypotheses

Several modern papers develop refined invariants and show partial preservation results: for topologically free continuous actions some cohomological or bounded‑cohomology invariants that detect amenability of the action are preserved under continuous orbit equivalence ^[4]. Other works relating cospectral radius, coamenability, and operator methods show techniques to transfer almost‑invariant vectors between coset spaces in the equivalence‑relation setting, again giving conditions when amenability‑type properties pass along inclusions or equivalences ^[5]. These results imply that amenability can be invariant under orbit equivalence in specific frameworks, but the general question has negative instances unless extra hypotheses are imposed ^{[4] [5]}.

5. Groupoid and inner‑amenability subtleties

The groupoid perspective clarifies distinctions: inner amenability for groupoids (or inner amenability of groups producing groupoids) and properties such as property Gamma or stability interact subtly with orbit equivalence. For example, free ergodic compact actions of an inner‑amenable group can produce an inner‑amenable orbit relation, but the transfer of other properties (Schmidt property, stability) from one equivalence relation to another under orbit inclusion may fail in general ^{[3] [6]}. Thus amenability‑type notions tied to central sequences or inner amenability require care when passing under orbit equivalence.

6. Takeaway and practical rule of thumb

If you start from an amenable group acting in the usual p.m.p. way, expect the orbit relation to be amenable/hyperfinite ^[1]. But the converse is false in general: non‑amenable groups can produce amenable orbit relations ^{[2] [3]}. Modern research therefore focuses on (a) identifying additional hypotheses under which amenability is preserved by orbit or continuous orbit equivalence (cohomological/topological freeness conditions) and (b) producing counterexamples or constructions showing failure in the absence of such hypotheses ^{[4] [5]}.

Limitations and what the sources do not say: the provided results do not settle a single blanket theorem “amenability is (or is not) invariant under all forms of orbit equivalence.” Instead they supply positive invariance results under extra structural assumptions (topological or cohomological) and explicit constructions demonstrating breakdowns when those assumptions fail ^{[4] [5] [2]}. Available sources do not mention a single universal criterion that answers the query in all settings.