Amenability versus hyperfiniteness for countable Borel equivalence relations

The problem of whether every amenable countable Borel equivalence relation is hyperfinite — the Borel analogue of the Connes–Feldman–Weiss theorem for measured equivalence relations — has been central to descriptive set theory since Weiss's 1984 question for actions of countable amenable groups and its general formulation by Kechris and by Jackson–Kechris–Louveau. Two implications remain open: whether amenability implies hyperfiniteness, and whether measure-hyperfiniteness (hyperfiniteness $\mu$-almost everywhere for every Borel probability measure $\mu$) implies hyperfiniteness as a Borel relation. By Connes–Feldman–Weiss, a positive answer to the second implies a positive answer to the first, and a single amenable non-hyperfinite counterexample — necessarily measure-hyperfinite — would refute both at once. The problem is open.

Background

Countable Borel equivalence relations (CBERs) are the central objects of one of the main research programs of modern descriptive set theory. A standard Borel space is a set carrying the Borel sigma-algebra of some Polish (separable, completely metrizable) topology, such as the real line or Cantor space $2^{\mathbb{N}}$. An equivalence relation $E$ on such a space is a CBER if it is a Borel subset of $X\times X$ and every class $[x]_E$ is countable. By a classical theorem of Feldman and Moore, every CBER is the orbit equivalence relation $E_G^X$ of a Borel action of some countable group $G$, where $xE_G^Xy$ iff $gx=y$ for some $g\in G$, so the theory of CBERs is also the definable theory of countable group actions.

A CBER $E$ is hyperfinite if it is the increasing union $E=\bigcup_n F_n$ of finite Borel equivalence relations, each $F_n\subseteq E$ Borel with finite classes; equivalently (Dougherty–Jackson–Kechris) $E$ is the orbit relation of a single Borel $\mathbb{Z}$-action, and equivalently $E$ is hypersmooth, meaning an increasing union of smooth Borel equivalence relations, where smooth means that the classes are exactly the fibers of a Borel map into a standard Borel space, so that they can be separated by countably many Borel invariants. Hyperfinite CBERs have a rigid structure: each is either smooth or Borel bireducible with the single canonical example $E_0$, eventual equality of binary sequences ($xE_0y$ iff $x(n)=y(n)$ for all but finitely many $n$) — where a Borel reduction of $E$ to $F$ is a Borel map $f$ with $xEy$ if and only if $f(x)Ff(y)$, and two relations are Borel bireducible when reductions exist in both directions. Determining which naturally occurring CBERs are hyperfinite has been a leading theme of the subject since the 1980s.

A CBER $E$ on $X$ is amenable (in the sense of Jackson–Kechris–Louveau) if there are Borel maps $\lambda_n:E\to[0,\infty)$ such that each $\lambda_{n,x}:=\lambda_n(x,\cdot)$ is a probability measure on the class $[x]_E$ and $\|\lambda_{n,x}-\lambda_{n,y}\|_1\to 0$ whenever $xEy$: the class measures become asymptotically invariant along the relation. And $E$ is measure-hyperfinite if for every Borel probability measure $\mu$ on $X$ there is a Borel set $A$ with $\mu(A)=1$ such that the restriction $E\upharpoonright A$ is hyperfinite.

Three implications are known. Hyperfiniteness trivially implies measure-hyperfiniteness, and it implies amenability. The deep measured-world theorem of Connes–Feldman–Weiss (1981) — an amenable measured equivalence relation is generated by a single transformation, hence hyperfinite almost everywhere — entails that every amenable CBER is measure-hyperfinite. The two remaining implications, from amenability to hyperfiniteness and from measure-hyperfiniteness to hyperfiniteness, are open; closing this gap is the problem below. For orbit relations of Borel actions of countable amenable groups (which are necessarily amenable relations) the question traces back to Weiss (1984) and is often called Weiss's question; it has positive answers for $\mathbb{Z}^d$ (Weiss), finitely generated groups of polynomial volume growth (Jackson–Kechris–Louveau), locally nilpotent groups (Schneider–Seward), and all countable abelian groups (Gao–Jackson). The breakthrough of Conley–Jackson–Marks–Seward–Tucker-Drob introduced Borel asymptotic dimension — a Borel analogue of Gromov's coarse asymptotic dimension: for a Borel extended metric $\rho$ on $X$ (a metric allowed to take the value $+\infty$) whose finite-distance relation is exactly $E$, it is the least $d$ such that for every radius $r$ there is a Borel equivalence relation $F\subseteq E$ whose classes have uniformly bounded finite diameter and such that every radius-$r$ ball meets at most $d+1$ classes of $F$ — proved that CBERs of finite Borel asymptotic dimension are hyperfinite, and thereby settled the question for polycyclic groups and, for free actions, for a large class of groups including the lamplighter group $\mathbb{Z}_2\wr\mathbb{Z}$ and all virtually solvable groups of finite Prüfer rank (a uniform finite bound on the number of generators of finitely generated subgroups), the first groups of exponential volume growth covered. Every known positive result requires hypotheses beyond amenability itself.

The source paper of Naryshkin–Vaccaro contributes the strongest known results in the treeable direction. A CBER is treeable if it admits an acyclic Borel graphing, i.e. a Borel graph whose connected components are exactly the $E$-classes and which contains no cycle; hyperfinite relations are treeable, and Jackson–Kechris–Louveau asked whether every amenable treeable CBER is hyperfinite. Naryshkin–Vaccaro define u-amenability — amenability in which the convergence $\|\lambda_{n,x}-\lambda_{n,y}\|_1\to 0$ is additionally uniform in the distance between $x$ and $y$ with respect to a Borel extended metric, typically the path metric of a graphing — and hyper-u-amenability, defined by requiring that the complete graphing of $E$ (the graph on $X$ in which any two $E$-equivalent points are adjacent) be an increasing union of Borel graphs each generating a u-amenable relation; their Proposition 3.5 shows the property is independent of the chosen graphing, hence intrinsic to $E$. Their main theorem states that every treeable hyper-u-amenable CBER is hyperfinite. Consequences: a measure-hyperfinite CBER that is the orbit relation of a free continuous action of a countable virtually free group (one containing a free subgroup of finite index) on a sigma-compact Polish space is hyperfinite; the treeable case of Weiss's question has a positive answer; and a treeable, amenable, Borel bounded CBER is hyperfinite, where Borel bounded (Boykin–Jackson) means that for every Borel $\varphi:X\to\mathbb{N}^{\mathbb{N}}$ there is a Borel $\psi:X\to\mathbb{N}^{\mathbb{N}}$ with $\varphi(x)\leq^{*}\psi(x)$ everywhere (eventual domination) and $\psi(x)=^{*}\psi(y)$ whenever $xEy$. In a parallel development, Elek and Timár proved that for bounded-degree Borel graphs a uniform version of amenability is equivalent to a randomized (probabilistic) version of hyperfiniteness, and that uniformly Borel amenable graphs are hyperfinite modulo a compressible invariant set — one admitting a Borel injection that moves every equivalence class properly into itself; such a set supports no invariant probability measure. Most recently, Grebík–Marks–Rozhoň–Shinko proved that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^{\gamma}))$, where $\gamma\approx 0.1523$ is the real root of $(1-\gamma)^3=4\gamma$, is hyperfinite, which gives a positive answer to Weiss's question for countable amenable groups locally of such volume growth, and they closed with conjectures linking Borel amenability, subexponential dimension growth, and hyperfiniteness that would imply a positive answer to Weiss's question. Kechris's 2024 monograph surveys this landscape and its open challenges.

Problem Statement

Settle, for countable Borel equivalence relations on standard Borel spaces in full generality — with no auxiliary hypotheses such as treeability, bounded degree, finite Borel asymptotic dimension, almost-everywhere weakening of the conclusion, or restriction to actions of particular classes of groups — the relationship between amenability, measure-hyperfiniteness, and hyperfiniteness. Concretely, the two open implications are:

(1) Amenability versus hyperfiniteness. Is every amenable countable Borel equivalence relation (amenable in the sense of Jackson–Kechris–Louveau: Borel maps $\lambda_n$ from $E$ to $[0,\infty)$ such that each $\lambda_{n,x}:=\lambda_n(x,\cdot)$ is a probability measure on the class $[x]_E$ and $\|\lambda_{n,x}-\lambda_{n,y}\|_1\to 0$ whenever $xEy$) hyperfinite, i.e. an increasing union of finite Borel equivalence relations on all of the underlying space?

(2) Measure-hyperfiniteness versus hyperfiniteness. If a countable Borel equivalence relation $E$ on a standard Borel space $X$ has hyperfinite restriction to some $\mu$-conull Borel set for every Borel probability measure $\mu$ on $X$, must $E$ itself be hyperfinite as a Borel relation on all of $X$?

These are the two faces of one question — whether the three properties coincide for countable Borel equivalence relations. The known implications are hyperfinite implies amenable, hyperfinite implies measure-hyperfinite, and (Connes–Feldman–Weiss) amenable implies measure-hyperfinite. The two implications are therefore equivalent to asking which of the following configurations holds: (a) all three properties coincide; (b) amenability and hyperfiniteness are equivalent, but there exists a non-amenable relation that is measure-hyperfinite and not hyperfinite; or (c) there exists an amenable relation that is not hyperfinite — which, by Connes–Feldman–Weiss, is automatically measure-hyperfinite, so a single counterexample of this kind refutes both implications at once.

References

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