Borel reducibility of the free Bernoulli shift of $F_2$ to orbit equivalence relations of tree boundary actions

Elayavalli, Oyakawa, Shinko and Spaas ('Hyperfiniteness for group actions on trees', arXiv:2307.10964, published in Proc. Amer. Math. Soc. 152 (2024)) proved hyperfiniteness results for orbit equivalence relations of boundary actions of countable groups on countable trees and posed (Question 5.2 in the arXiv versions, 2023 and 2025) the question of whether Hjorth's theorem -- the free part of the Bernoulli shift $F_2 \curvearrowright 2^{F_2}$ does not Borel-reduce to the orbit equivalence relation of any prodiscrete (equivalently, rooted-tree boundary) action -- extends to boundary actions of arbitrary actions of countable groups on countable trees. The question is open as of August 2026.

Background

Descriptive set theory studies countable Borel equivalence relations (CBERs): equivalence relations $E$ on a standard Borel space that are Borel as sets of pairs and have countable classes. Every CBER is the orbit equivalence relation of a Borel action of a countable group, and complexity is compared by Borel reduction: $E$ Borel reduces to $F$, written $E \le_B F$, if there is a Borel map $f$ with $x \, E \, x'$ if and only if $f(x) \, F \, f(x')$. The simplest aperiodic CBERs (those with all classes infinite) are the hyperfinite ones, that is, increasing unions of finite Borel equivalence relations; a major program determines hyperfiniteness for orbit equivalence relations of natural group actions (the general question of whether every Borel action of a countable amenable group (a group admitting a finitely additive, left-invariant probability measure on the set of its subsets) is hyperfinite remains open, with substantial partial results by Conley, Jackson, Marks, Seward and Tucker-Drob).

Two objects anchor this problem. First, the Bernoulli shift of a countable group $\Gamma$ is the action of $\Gamma$ by left translation on $2^{\Gamma}$, the compact space of $\{0,1\}$-colorings of $\Gamma$; its free part is the restriction of the action (and of its orbit equivalence relation) to the invariant Borel set of colorings with trivial stabilizer. By a classical result of Jackson, Kechris and Louveau, the orbit equivalence relation of any probability-measure-preserving (pmp), almost-everywhere-free (free off a null set) action of a non-amenable group is not measure-hyperfinite, i.e. not hyperfinite on any conull invariant Borel set for the invariant measure; applied to the Bernoulli shift of the free group $F_2$ with product measure, the free part of the shift is not hyperfinite, and it serves as the canonical non-hyperfinite benchmark for how complicated an orbit equivalence relation can be. Second, if a countable group $G$ acts by automorphisms on a countable tree $T$ (a connected acyclic graph with countably many vertices), the action induces a Borel action of $G$ on the Gromov boundary $\partial T$, the space of ends of $T$ -- equivalence classes of one-sided infinite geodesic rays, two rays being identified when they eventually coincide; $E_G^{\partial T}$ denotes the resulting orbit equivalence relation.

A theorem of Hjorth ('A converse to Dye's theorem', 2005) shows that prodiscrete actions -- Borel actions that are inverse limits of actions on countable sets, which by a fact of Kechris are exactly the boundary actions of rooted tree actions (actions fixing a vertex) -- are never complicated enough to receive the free Bernoulli shift: the free part of the Bernoulli shift $F_2 \curvearrowright 2^{F_2}$ does not Borel-reduce to the orbit equivalence relation of any prodiscrete action. In Kechris's terminology the shift is antimodular: there is not even a countable-to-one Borel homomorphism (a Borel map carrying equivalent points to equivalent points, with countable fibers) from its orbit equivalence relation to that of any modular (prodiscrete) action, and this result is how Hjorth first produced more than two treeable CBERs up to Borel reducibility (a CBER is treeable if it admits a Borel acyclic graphing, i.e. can be generated by a Borel forest). Kechris and later Ioana extended antimodularity to further classes of probability-measure-preserving actions via representation-theoretic and spectral-gap conditions, and Kechris's 2024 monograph treats the topic in its Section 10.8.

In parallel, the boundary-action hyperfiniteness program grew: the tail-equivalence theorem of Dougherty, Jackson and Kechris (hyperfiniteness of the relation of eventual agreement between one-sided infinite binary sequences) implies that the action of $F_2$ on the boundary of its Cayley tree is hyperfinite; Huang, Sabok and Shinko, Marquis and Sabok, and Oyakawa proved hyperfiniteness for boundary actions of cubulated and general hyperbolic groups, relatively hyperbolic groups, and acylindrically hyperbolic groups. Elayavalli, Oyakawa, Shinko and Spaas identified a stabilizer condition on tree actions (every geodesic ray has an initial segment whose pointwise stabilizer equals that of the ray; satisfied for instance by acylindrical actions, where stabilizers of far-apart vertex pairs are uniformly bounded) implying that $E_G^{\partial T}$ is Borel hyperfinite, a uniformly coamenable weakening implying measure-hyperfiniteness, and, crucially for this problem, they documented boundary actions that are not hyperfinite (free pmp prodiscrete actions of residually finite groups, i.e. groups whose nontrivial elements each survive in some finite quotient). Hyperfiniteness therefore cannot settle how complicated tree-boundary orbit equivalence relations can be, and whether the class of such equivalence relations beyond the rooted case can contain the free Bernoulli shift is a genuine open question, posed by Elayavalli, Oyakawa, Shinko and Spaas as Question 5.2 of 'Hyperfiniteness for group actions on trees'.

Problem Statement

Let $F_2 \curvearrowright 2^{F_2}$ be the Bernoulli shift of the free group on two generators, and let $E^\ast$ be its orbit equivalence relation restricted to the free part (the invariant Borel set of colorings with trivial stabilizer). Question (Elayavalli, Oyakawa, Shinko and Spaas, 'Hyperfiniteness for group actions on trees', arXiv:2307.10964, Proc. Amer. Math. Soc. 152 (2024); Question 5.2 in the arXiv versions): is it true that for every action $G \curvearrowright T$ of a countable group $G$ by automorphisms on a countable tree $T$, with $E_G^{\partial T}$ the orbit equivalence relation of the induced action of $G$ on the Gromov boundary $\partial T$ of $T$ (its space of ends: equivalence classes of one-sided infinite geodesic rays in $T$, two rays being identified when they eventually coincide), the free part of the Bernoulli shift does not Borel reduce to $E_G^{\partial T}$, i.e. there is no Borel map $f$ from the free part of $2^{F_2}$ to $\partial T$ with $x \, E^\ast \, x'$ if and only if $f(x) \, E_G^{\partial T} \, f(x')$? Equivalently: does there exist an action of a countable group on a countable tree whose boundary orbit equivalence relation is a Borel-reduction target for the free Bernoulli shift of $F_2$?

References

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