Hyperfiniteness of generic Cantor actions of exact groups

For a countable discrete group that is locally of finite asymptotic dimension, Iyer and Shinko proved that the generic continuous action on Cantor space has hyperfinite orbit equivalence relation, answering a question of Frisch–Kechris–Shinko–Vidnyánszky for free groups. The underlying question is open in general: if a countable group $\Gamma$ is exact (its reduced group C*-algebra is exact), is the set of hyperfinite Cantor actions comeager in the Polish space $\mathrm{Act}(\Gamma)$ of all continuous actions of $\Gamma$ on Cantor space? This problem asks for a full resolution — either a proof that hyperfinite Cantor actions are comeager for every countable exact group, or an explicit countable exact group for which they are not. The representative instance not reached by any current technique is the solvable group $\mathbb{Z}\wr\mathbb{Z}$, which is amenable, hence exact, but not locally of finite asymptotic dimension. A negative answer for any exact group would simultaneously refute the conjecture that every measure-hyperfinite countable Borel equivalence relation is hyperfinite (Kechris, Problem 17.7).

Background

A countable Borel equivalence relation (CBER) is an equivalence relation $E$ on a standard Borel space $X$ which is Borel as a subset of $X\times X$ and has countable classes. CBERs are compared by Borel reducibility: $E\leq_{B}F$ if some Borel map $f$ satisfies $x\,E\,x'\iff f(x)\,F\,f(x')$. The simplest CBERs are the smooth ones, those reducible to equality; a classical dichotomy (see Kechris 2024, Theorem 6.5) shows every non-smooth CBER contains a copy of the relation $E_{0}$ of eventual equality of binary sequences. A CBER is hyperfinite if it Borel reduces to $E_{0}$, equivalently (by the structure theory of Dougherty–Jackson–Kechris) if it is an increasing union of Borel equivalence relations with finite classes. Hyperfinite CBERs are the minimal non-smooth objects of the hierarchy and the baseline against which all other CBERs are measured.

Hyperfiniteness is tightly connected to amenability. By the Connes–Feldman–Weiss theorem, every orbit equivalence relation of a countable amenable group is measure-hyperfinite: for every Borel probability measure $\mu$ there is a Borel set of $\mu$-measure one on which the relation is hyperfinite. Weiss asked in 1984 whether the measure qualification can be removed — is every orbit equivalence relation of every countable amenable group hyperfinite? This is Problem 17.8 of Kechris's 2024 monograph and remains open, even for solvable groups; it is known positively for all Borel actions of polycyclic groups and groups of local polynomial volume-growth, and for all free Borel actions of a class of solvable groups including the lamplighter $\mathbb{Z}_{2}\wr\mathbb{Z}$ (Conley–Jackson–Marks–Seward–Tucker-Drob). Problem 17.7 of the same monograph asks the intermediate question: is every measure-hyperfinite CBER hyperfinite? A positive answer would settle Weiss's question; a negative answer is not excluded and could take a strong form, since measure-hyperfiniteness is a coanalytic ($\mathbf{\Pi}^{1}_{1}$) property while hyperfiniteness could be as complex as $\mathbf{\Sigma}^{1}_{2}$-complete, which would yield many measure-hyperfinite CBERs that are not hyperfinite (Dougherty–Jackson–Kechris).

The present problem is the Baire-category analogue of this circle of questions. A Cantor action of a countable discrete group $\Gamma$ is a continuous action of $\Gamma$ on the Cantor space $2^{\mathbb{N}}$, that is, a homomorphism $\Gamma\to\mathrm{Homeo}(2^{\mathbb{N}})$; the space $\mathrm{Act}(\Gamma)$ of Cantor actions is a Polish space, a closed subspace of $\mathrm{Homeo}(2^{\mathbb{N}})^{\Gamma}$ with the compact-open topology. A subset of a Polish space is comeager if it contains a countable intersection of dense open sets, and the 'generic' action is one lying in a fixed comeager set; call an action (measure-)hyperfinite when its orbit equivalence relation is. Suzuki proved that if $\Gamma$ is exact — the C*-algebraic condition that the reduced group C*-algebra $C^{*}_{r}(\Gamma)$ is exact, satisfied by every amenable group and by free groups — then the measure-hyperfinite Cantor actions of $\Gamma$ form a comeager set. Frisch–Kechris–Shinko–Vidnyánszky (Problem 8.0.16, stated for spaces of subshifts; the action-space form is equivalent by a result of Hochman recorded there) and Iyer–Shinko (Question 2) ask whether hyperfiniteness itself is generic in this sense for exact groups, noting that a negative answer would immediately give a negative answer to Problem 17.7.

Iyer–Shinko answered this positively for groups locally of finite asymptotic dimension. Asymptotic dimension is Gromov's coarse invariant: $\mathrm{asdim}(\Gamma)$ is the least $n$ such that at every scale the Cayley graph of $\Gamma$ admits an $(n+1)$-coloring whose monochromatic components are uniformly bounded; a group is locally of finite asymptotic dimension when all of its finitely generated subgroups have finite asymptotic dimension (free groups, hyperbolic groups, and mapping class groups of finite-type surfaces all qualify). Conley–Jackson–Marks–Seward–Tucker-Drob introduced the Borel counterpart for a Borel graphing — Borel colorings whose monochromatic components are uniformly bounded — and proved that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite. Iyer–Shinko show that the generic Cantor action $\mathbf{a}$ satisfies $\mathrm{asdim}_{c}(\mathbf{a}\upharpoonright\Delta)=\mathrm{asdim}(\Delta)$ for every finitely generated subgroup $\Delta\leq\Gamma$ (where $\mathrm{asdim}_{c}$ uses continuous colorings), whence groups locally of finite asymptotic dimension have generic hyperfinite Cantor actions. This covers amenable examples such as $((\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z})^{2}$ — for which, however, it is still open whether all, rather than generic, Cantor actions are hyperfinite — but it does not reach the solvable group $\mathbb{Z}\wr\mathbb{Z}=(\bigoplus_{\mathbb{Z}}\mathbb{Z})\rtimes\mathbb{Z}$, which is amenable and hence exact, yet contains $\mathbb{Z}^{n}$ for every $n$ in its lamp group, so it is not locally of finite asymptotic dimension and even the generic Cantor action of $\mathbb{Z}\wr\mathbb{Z}$ is not known to be hyperfinite. Wreath products of this kind are the acknowledged frontier: the most recent work on the adjacent Weiss problem (Grebík–Marks–Rozhoň–Shinko, August 2026) identifies $\mathbb{Z}\wr\mathbb{Z}$ as its key open test case.

Problem Statement

Let $\Gamma$ be a countable discrete group which is exact, i.e. its reduced group C*-algebra $C^{*}_{r}(\Gamma)$ is exact. Let $\mathrm{Act}(\Gamma)$ denote the Polish space of continuous actions of $\Gamma$ on the Cantor space $2^{\mathbb{N}}$ (homomorphisms $\Gamma\to\mathrm{Homeo}(2^{\mathbb{N}})$ with the compact-open topology), and call an action hyperfinite when its orbit equivalence relation is a hyperfinite countable Borel equivalence relation, i.e. an increasing union of Borel equivalence relations with finite classes. A subset of a Polish space is comeager if it contains a countable intersection of dense open sets.

Question (Iyer–Shinko, Question 2; Frisch–Kechris–Shinko–Vidnyánszky, Problem 8.0.16): is the set

$\{\mathbf{a}\in\mathrm{Act}(\Gamma):\ \mathbf{a}\ \text{is hyperfinite}\}$

comeager in $\mathrm{Act}(\Gamma)$ for every countable exact group $\Gamma$?

The problem is to resolve this question in full generality: either prove that for every countable exact group $\Gamma$ the hyperfinite Cantor actions are comeager in $\mathrm{Act}(\Gamma)$, or exhibit a countable exact group $\Gamma$ for which they are not comeager. The instance highlighted by the sources as beyond all current methods is the solvable group $\mathbb{Z}\wr\mathbb{Z}=(\bigoplus_{\mathbb{Z}}\mathbb{Z})\rtimes\mathbb{Z}$, which is amenable and hence exact but is not locally of finite asymptotic dimension (i.e. not all of its finitely generated subgroups have finite asymptotic dimension); even the generic Cantor action of $\mathbb{Z}\wr\mathbb{Z}$ is not currently known to be hyperfinite. Note that a negative answer for any exact group would simultaneously give a negative answer to the question of whether every measure-hyperfinite countable Borel equivalence relation is hyperfinite (Kechris, Problem 17.7) — a countable Borel equivalence relation is measure-hyperfinite if it is hyperfinite on a Borel set of full measure for every Borel probability measure — since Suzuki's theorem makes measure-hyperfinite actions comeager for exact groups.

References

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