Hyperfiniteness of bounded-to-one actions of commutative monoids

Abstract: A theorem of Dougherty--Jackson--Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao--Jackson on Borel actions of countable abelian groups.

• The paper proves that every bounded-to-one Borel action of a countable commutative monoid generates a hyperfinite orbit equivalence relation, extending results for abelian groups and tail equivalence.
• Its methodology combines Borel asymptotic dimension, a monoid-to-group transfer principle for local algorithms, Rédei’s Noetherian stable-point reduction, and quantitative dimension-growth estimates.
• The result covers both finitely generated and infinitely generated commutative monoids while leaving unrestricted actions, finite-dimensionality without freeness, and the general union problem as open directions.

Context and motivation

The paper studies hyperfiniteness of orbit equivalence relations of Borel actions of countable commutative monoids. For a Borel action $M \curvearrowright X$ of a countable monoid on a standard Borel space, the orbit equivalence relation $E_M^X$ is the smallest equivalence relation with $x \mathrel{E} mx$; when $M$ is commutative this is characterized by the existence of $m, n \in M$ with $mx = ny$. The central open problem, a folklore generalization of Weiss's question for amenable groups and of Gao–Jackson's theorem for abelian groups [GJ15], asks whether $E_M^X$ is hypersmooth for every countable commutative monoid action. The paper gives an affirmative answer under a bounded-to-one hypothesis: each monoid element acts by a map whose fibers have uniformly bounded (but element-dependent) size.

The result subsumes two classical theorems. First, since group actions are one-to-one in each coordinate, it recovers Gao–Jackson's theorem that orbit equivalence relations of countable abelian groups are hyperfinite. Second, it extends the Dougherty–Jackson–Kechris theorem that tail equivalence of a single Borel function is hypersmooth [DJK94], which corresponds to the case $M = \N$; even the case of two commuting bounded-to-one maps was previously open.

Main theorem

Theorem. If $M$ is a countable commutative monoid acting boundedly-to-one and in a Borel fashion on a standard Borel space $X$, then $E_M^X$0 is hyperfinite.

The proof strategy follows the now-standard route through Borel asymptotic dimension, introduced by Conley–Jackson–Marks–Seward–Tucker-Drob [CJMST23]: finite Borel asymptotic dimension of a locally finite Borel graph implies its connectedness relation is hyperfinite. Two obstructions must be overcome: non-freeness of the action, and non-finite-generation of the monoid.

Reduction to free actions via Rédei's theorem

For finitely generated commutative monoids, freeness is handled using Rédei's theorem that such monoids are Noetherian (every congruence is finitely generated). For each point $E_M^X$1, the relation $E_M^X$2 defined by $E_M^X$3 is a congruence, and along an orbit these congruences form a directed system under inclusion. Noetherianity yields maximal elements; the set $E_M^X$4 of stable points (where $E_M^X$5 is maximal) is an $E_M^X$6-invariant complete section, and the action descends to a free action of the quotient monoid $E_M^X$7 on $E_M^X$8. Since a CBER restricted to a complete section being hyperfinite implies the whole relation is hyperfinite, this reduces the finitely generated case to free actions. The authors note this reduction appeared independently, with essentially the same proof, in a preprint of Wang.

Local algorithms and the monoid–group transfer principle

A substantial technical contribution is a development of Linial's deterministic $E_M^X$9 model for monoid actions, building on Bernshteyn's dictionary between local algorithms and descriptive combinatorics [Ber23a, Ber23b]. Round complexity is defined via local reductions from an LCL to injective-labeling problems $x \mathrel{E} mx$0, and a quantitative transfer lemma shows that if $x \mathrel{E} mx$1, then every Borel action where generators act at most $x \mathrel{E} mx$2-to-one admits a Borel $x \mathrel{E} mx$3-labeling on the $x \mathrel{E} mx$4-free points, with $x \mathrel{E} mx$5 — the bounded-to-one hypothesis supplies the needed Borel proper colorings via Kechris–Solecki–Todorcevic [KST99].

The key structural result is a transfer theorem: for a finitely generated cancellative commutative monoid $x \mathrel{E} mx$6 with group completion $x \mathrel{E} mx$7, and any LCL $x \mathrel{E} mx$8 on $x \mathrel{E} mx$9, the following are equivalent: continuous solvability on free continuous actions of $M$0 on zero-dimensional Polish spaces; continuous solvability on free clopen-preserving bounded-to-one actions of $M$1; membership in $M$2; and membership in $M$3. Consequently, every free Borel bounded-to-one $M$4-action admits a Borel $M$5-labeling whenever any of these hold. The proof routes through round-complexity bounds showing $M$6 using lcm arguments. This yields:

Theorem. Free bounded-to-one Borel actions of finitely generated commutative monoids satisfy $M$7 (indeed $M$8).

The authors emphasize that they could not remove the freeness assumption here at the level of asymptotic dimension itself — a point revisited below.

Smooth separation index

As an independent contribution, the paper introduces the Borel asymptotic smooth separation index $M$9, weakening the asymptotic separation index of [CJMST23] by requiring only smoothness rather than finite classes of witnesses. Finite values coincide ($m, n \in M$0), but $m, n \in M$1 behaves better under pullback along countable-to-one bornologous maps — a property $m, n \in M$2 lacks. Combining this with the stable-point machinery gives $m, n \in M$3 for all bounded-to-one Borel actions of finitely generated commutative monoids, whence consequences such as: if the Schreier graph $m, n \in M$4 has a $m, n \in M$5-coloring then it has a Borel $m, n \in M$6-coloring; in particular, Schreier graphs of free bounded-to-one Borel $m, n \in M$7-actions admit Borel 3-colorings. A corollary worth highlighting is the equality $m, n \in M$8 for finitely generated commutative monoid actions.

Quantitative refinements and the general case

Non-finitely-generated commutative monoids are not Noetherian (e.g., $m, n \in M$9), so the stable-point trick fails verbatim, and the "union problem" — whether increasing unions of hyperfinite CBERs are hyperfinite — remains open in general. The paper instead proves a quantitative weakening sufficient for the union theorem of Grebík–Marks–Rozhoň–Weilacher [GMRS26].

Writing $mx = ny$0 as a quotient of $mx = ny$1 and decomposing into Schreier graphs $mx = ny$2 of $mx = ny$3-actions, the goal is $mx = ny$4-dimension growth for each $mx = ny$5. Two ingredients combine:

• Quantitative dimension witnesses. Via a doubling lemma for balls in abelian groups and Linial's MIS algorithm, the dimension-witnessing LCL satisfies $mx = ny$6 with $mx = ny$7, $mx = ny$8, uniformly over quotients $mx = ny$9.
• Local stability. A refined notion of $E_M^X$0-stability for $E_M^X$1-sets, built on a variant of Steinitz's lemma (the "dog walker" argument), shows every point can be moved to an $E_M^X$2-stable point within $E_M^X$3 steps, that the associated subgroup chains have length $E_M^X$4, and that the map to stable points is Lipschitz in an appropriate sense.

Combining these, each $E_M^X$5 admits Borel uncolored dimension $E_M^X$6 witnesses at scale $E_M^X$7 with polynomial radius: the image of a ball meets only $E_M^X$8 many stabilizer-congruence strata, and within each stratum the transferred local algorithm compresses balls to $E_M^X$9 points. A general colored/uncolored conversion for $M = \N$0-dimension growth, together with the union criterion of [GMRS26], then yields hyperfiniteness of $M = \N$1.

Limitations and open questions

Several gaps remain explicit. The freeness hypothesis cannot currently be removed from the finite-asymptotic-dimension statement: whether every bounded-to-one $M = \N$2-set for a finitely generated commutative monoid $M = \N$3 has finite classical asymptotic dimension — and whether $M = \N$4 — is left open; a positive answer would give a cleaner proof of the main theorem via the equality of classical and Borel dimensions. More broadly, the motivating question for unrestricted (not necessarily bounded-to-one) actions of countable commutative monoids remains open, as does the general union problem for CBERs. It is also unknown whether any locally finite Borel graph has $M = \N$5. Finally, the authors note the higher-rank motivation: boundary actions of higher-rank hyperbolic-type settings involve finitely generated commutative monoids such as $M = \N$6, and the free-case dimension theorem is only a starting point for hyperfiniteness of those actions.

Conclusion

The paper establishes that bounded-to-one Borel actions of countable commutative monoids generate hyperfinite equivalence relations, unifying Gao–Jackson's abelian group theorem with the Dougherty–Jackson–Kechris tail-equivalence theorem. Methodologically, it contributes a monoid-valued $M = \N$7 theory with a transfer principle to abelian groups, a Noetherian stable-point decomposition handling non-freeness, and quantitative dimension-growth estimates sufficient to bypass both the union problem and the failure of Noetherianity in the infinitely generated case.