Hyper-u-Amenability in Borel Relations

• Hyper-u-amenability is a strengthened form of amenability for countable Borel equivalence relations that requires uniform convergence of approximations over metric-defined neighborhoods.
• It utilizes decompositions into finite-degree u-amenable subgraphs, employing acyclic graphings and controlled Borel asymptotic dimension to achieve hyperfiniteness.
• By linking amenability with treeability, hyper-u-amenability provides a powerful framework that transitions amenability-type hypotheses into strong structural finiteness properties.

Searching arXiv for papers on hyper-u-amenability and closely related terminology. Hyper-$u$-amenability is a notion for countable Borel equivalence relations introduced as a strengthened form of amenability that is implied by hyperfiniteness and designed to interact effectively with treeability and Borel asymptotic dimension. In the formulation introduced in "Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations" (Naryshkin et al., 10 Jul 2025), the concept is built from the metric-dependent notion of $u$-amenability and then promoted to an intrinsic closure property via graphings. Its central role is to bridge amenability-type hypotheses and hyperfiniteness: for treeable countable Borel equivalence relations, hyper-$u$-amenability implies hyperfiniteness (Naryshkin et al., 10 Jul 2025).

1. Definition and ambient framework

The theory is formulated for a standard Borel space $X$ and a countable Borel equivalence relation $E \subseteq X^2$, meaning that $E$ is a Borel subset of $X\times X$ and each equivalence class $[x]_E$ is countable (Naryshkin et al., 10 Jul 2025). A Borel graph $G=(X,R)$ is a symmetric, irreflexive Borel subset $R\subseteq X^2$, and its connectedness relation $u$0 is the equivalence relation of lying in the same connected component (Naryshkin et al., 10 Jul 2025).

The paper distinguishes several standard notions. A countable Borel equivalence relation is finite if every class is finite, hyperfinite if it is the increasing union of finite Borel equivalence relations, measure-hyperfinite if every Borel probability measure concentrates on a Borel set on which the restricted relation is hyperfinite, and treeable if it admits an acyclic graphing (Naryshkin et al., 10 Jul 2025). The development uses a Borel extended metric $u$1 on $u$2, often the shortest-path metric $u$3 induced by a graphing $u$4 (Naryshkin et al., 10 Jul 2025).

The starting point is the standard JKL notion of amenability for countable Borel equivalence relations. A relation $u$5 is amenable if there are Borel maps

$u$6

such that for each $u$7, the function $u$8 belongs to $u$9, satisfies $u$0, and obeys

$u$1

for all $u$2 (Naryshkin et al., 10 Jul 2025). This is the usual approximate-invariant-means formulation along equivalence classes.

2. $u$3-amenability and the passage to hyper-$u$4-amenability

The paper strengthens amenability by imposing uniformity relative to a metric. If $u$5 is a Borel extended metric space and $u$6, then $u$7 is $u$8-amenable with respect to $u$9 if there are Borel maps

$X$0

such that for each $X$1, $X$2, $X$3, and

$X$4

for every $X$5 (Naryshkin et al., 10 Jul 2025). A Borel graph $X$6 is $X$7-amenable if $X$8 is $X$9-amenable with respect to $E \subseteq X^2$0 (Naryshkin et al., 10 Jul 2025).

The distinction from ordinary amenability is exact. Amenability requires convergence for each fixed pair $E \subseteq X^2$1, whereas $E \subseteq X^2$2-amenability requires convergence uniformly over all pairs lying within any fixed $E \subseteq X^2$3-radius (Naryshkin et al., 10 Jul 2025). The paper emphasizes that this dependence on the metric is a feature, but also the reason $E \subseteq X^2$4-amenability is not purely a property of $E \subseteq X^2$5 unless it is repackaged into an intrinsic notion (Naryshkin et al., 10 Jul 2025).

Hyper-$E \subseteq X^2$6-amenability is that intrinsic notion. A countable Borel equivalence relation $E \subseteq X^2$7 is hyper-$E \subseteq X^2$8-amenable if it admits a graphing $E \subseteq X^2$9 that is an increasing union of Borel $E$0-amenable graphs,

$E$1

where each $E$2 is $E$3-amenable (Naryshkin et al., 10 Jul 2025). The paper proves that this does not depend on the initial graphing in a problematic way: if one graphing has such a decomposition, then every graphing does, and the pieces can be chosen of finite degree (Naryshkin et al., 10 Jul 2025). In the form recorded in the paper, Proposition 4.2 yields

$E$4

(Naryshkin et al., 10 Jul 2025).

3. Relation to amenability, hyperfiniteness, and treeability

The motivation is a standard asymmetry in the theory of countable Borel equivalence relations: hyperfinite implies amenable is known, while whether amenable implies hyperfinite is open in general (Naryshkin et al., 10 Jul 2025). Hyper-$E$5-amenability is introduced as a strong form of amenability still implied by hyperfiniteness, but strong enough to force finite Borel asymptotic dimension in acyclic finite-degree graphs (Naryshkin et al., 10 Jul 2025).

Treeability is structurally central. The main structural theorem applies to acyclic graphings, and hyperfinite relations are known to be treeable (Naryshkin et al., 10 Jul 2025). Against that background, the paper proves the converse under the stronger hypothesis of hyper-$E$6-amenability: $E$7 (Naryshkin et al., 10 Jul 2025). This identifies hyper-$E$8-amenability as a strengthening of amenability tailored to the treeable setting.

A plausible implication is that hyper-$E$9-amenability is intended not merely as an abstract strengthening, but as a uniformity condition that enables geometric control over treeings. The paper makes this precise through Borel asymptotic dimension rather than through purely measure-theoretic arguments (Naryshkin et al., 10 Jul 2025).

4. Geometric mechanism: acyclic graphs and Borel asymptotic dimension

The main technical theorem concerns acyclic finite-degree graphs. If $X\times X$0 is an acyclic Borel graph with $X\times X$1 and $X\times X$2 is $X\times X$3-amenable, then

$X\times X$4

In particular, $X\times X$5 is hyperfinite (Naryshkin et al., 10 Jul 2025). The paper identifies this as Theorem 5.2, also Theorem 1.1 in the body (Naryshkin et al., 10 Jul 2025).

The proof uses $X\times X$6-amenability to construct a partial Borel orientation and then invokes the Borel asymptotic dimension machinery of Conley–Jackson–Marks–Seward–Tucker-Drob (Naryshkin et al., 10 Jul 2025). Several ingredients are singled out.

For an acyclic graph $X\times X$7 and a Borel family $X\times X$8, the paper defines Borel maps

$X\times X$9

where $[x]_E$0 is obtained by deleting the edge $[x]_E$1 (Naryshkin et al., 10 Jul 2025). These maps quantify how much of $[x]_E$2 and $[x]_E$3 lies on each side of a cut edge.

A further ingredient is a quasi-oriented decomposition lemma, stated as Lemma 4.3, which gives a sufficient criterion for

$[x]_E$4

under a decomposition $[x]_E$5 satisfying orientation and sparsity conditions (Naryshkin et al., 10 Jul 2025). In the acyclic $[x]_E$6-amenable case, the hypotheses are verified sharply enough to conclude $[x]_E$7 (Naryshkin et al., 10 Jul 2025).

The paper also records the explicit $[x]_E$8-amenability estimate used in the construction: given $[x]_E$9, one chooses $G=(X,R)$0 so that

$G=(X,R)$1

then defines $G=(X,R)$2 by those edges across which the mass on the two sides is decisively biased (Naryshkin et al., 10 Jul 2025). This yields a partial orientation with out-degree $G=(X,R)$3, while the complementary subgraph $G=(X,R)$4 satisfies the sparse-leaf and separation properties required by the criterion (Naryshkin et al., 10 Jul 2025).

5. The main theorem for treeable equivalence relations

The headline theorem states that if $G=(X,R)$5 is a treeable and hyper-$G=(X,R)$6-amenable countable Borel equivalence relation, then $G=(X,R)$7 is hyperfinite (Naryshkin et al., 10 Jul 2025). The argument is concise once the geometric machinery is available.

One first chooses an acyclic graphing $G=(X,R)$8 of $G=(X,R)$9 (Naryshkin et al., 10 Jul 2025). By Proposition 4.2, $R\subseteq X^2$0 can be written as an increasing union $R\subseteq X^2$1 with each $R\subseteq X^2$2 $R\subseteq X^2$3-amenable and of finite degree (Naryshkin et al., 10 Jul 2025). Since each $R\subseteq X^2$4 is acyclic and $R\subseteq X^2$5-amenable, Theorem 5.2 implies that $R\subseteq X^2$6 has finite asymptotic dimension and hence each $R\subseteq X^2$7 is hyperfinite (Naryshkin et al., 10 Jul 2025). The paper then applies a union proposition in Borel asymptotic dimension: if $R\subseteq X^2$8 and each $R\subseteq X^2$9 has finite asymptotic dimension, then

$u$00

is hyperfinite (Naryshkin et al., 10 Jul 2025). Since $u$01, hyperfiniteness follows (Naryshkin et al., 10 Jul 2025).

This theorem is the main reason the notion was introduced. The paper explicitly presents hyper-$u$02-amenability as a strengthening of amenability designed so that the treeable case becomes tractable (Naryshkin et al., 10 Jul 2025).

6. Classes of examples and corollaries

The paper gives several classes in which hyper-$u$03-amenability occurs naturally (Naryshkin et al., 10 Jul 2025). The following summary collects the cases explicitly stated.

Setting	Conclusion
Continuous topologically amenable action $u$04	For every compact $u$05 and finite $u$06, the Schreier graph $u$07 is $u$08-amenable
Continuous action on a $u$09-compact Polish space, all stabilizers amenable, and $u$10 measure-hyperfinite	$u$11 is hyper-$u$12-amenable
Borel action of a countable amenable group	The orbit equivalence relation is hyper-$u$13-amenable
Amenable and Borel bounded equivalence relation	It is hyper-$u$14-amenable

The corollaries emphasize the interaction with treeability. If $u$15 with $u$16 or $u$17 acts freely and continuously on a $u$18-compact Polish space $u$19, and the orbit relation $u$20 is measure-hyperfinite, then $u$21 is hyperfinite (Naryshkin et al., 10 Jul 2025). The route is that such relations are hyper-$u$22-amenable, and the main theorem then applies (Naryshkin et al., 10 Jul 2025).

If a countable amenable group acts Borelly on a standard Borel space $u$23 and the orbit equivalence relation $u$24 is treeable, then $u$25 is hyperfinite (Naryshkin et al., 10 Jul 2025). The paper attributes the hyper-$u$26-amenability input to Proposition 4.3 and then applies treeable $u$27 hyper-$u$28-amenable $u$29 hyperfinite (Naryshkin et al., 10 Jul 2025).

A further corollary states that if $u$30 is treeable, amenable, and Borel bounded, then $u$31 is hyperfinite (Naryshkin et al., 10 Jul 2025). Here Borel boundedness means that for every Borel map $u$32, there exists a Borel $u$33 such that $u$34 and $u$35 whenever $u$36 (Naryshkin et al., 10 Jul 2025). The paper shows that amenable $u$37 Borel bounded implies hyper-$u$38-amenable, so the treeable theorem again yields hyperfiniteness (Naryshkin et al., 10 Jul 2025).

7. Terminology, scope, and nearby notions

The term hyper-$u$39-amenability is used formally in the setting of countable Borel equivalence relations in (Naryshkin et al., 10 Jul 2025). The paper explicitly distinguishes it from $u$40-amenability: the latter depends on a chosen metric or graphing, whereas hyper-$u$41-amenability is introduced as the intrinsic graphing-independent closure property (Naryshkin et al., 10 Jul 2025).

The literature also contains nearby amenability notions with similar names but different meanings. "Operator ultra-amenability" for completely contractive Banach algebras is defined by the condition that every ultrapower $u$42 is operator amenable (Forrest et al., 2016). For Fourier algebras, that notion imposes severe restrictions: if $u$43 is operator ultra-amenable, then $u$44 is discrete and amenable and contains no infinite abelian subgroup (Forrest et al., 2016). The terminology is adjacent, but it belongs to operator space theory rather than Borel equivalence relations.

Likewise, work on amenability of unitary co-representations of locally compact quantum groups studies left- and right-invariant means for co-representations and proves that amenability passes under weak containment (Ng et al., 2016). That theory does not define hyper-$u$45-amenability as a separate notion (Ng et al., 2016). In a different direction, extreme amenability of the unitary group of the hyperfinite II$u$46-factor is established via Lévy concentration, again without introducing hyper-$u$47-amenability (Dowerk et al., 2015). These neighboring uses of amenability language are relevant for disambiguation, but they are conceptually distinct from the Borel-equivalence-relation notion introduced in 2025 (Naryshkin et al., 10 Jul 2025).

The central conceptual takeaway of (Naryshkin et al., 10 Jul 2025) is that hyper-$u$48-amenability functions as a stronger uniform version of amenability that is still satisfied in many natural situations, yet is strong enough to control the geometry of treeings through Borel asymptotic dimension. The paper summarizes this mechanism by the implications

$u$49

and then

$u$50

(Naryshkin et al., 10 Jul 2025). This suggests that the notion is best understood as a bridge between amenability-type hypotheses and structural finiteness properties in the treeable regime.