Generic Cantor actions of Z wr Z are hyperfinite

Abstract

We give an affirmative answer to Question 3 in Iyer's contribution, joint with Shinko, to Oberwolfach Report 2/2025. A dense Gδ of continuous actions of the restricted wreath product ℤ ≀ ℤ on Cantor space has hyperfinite orbit equivalence relation on the entire space. We specialize established Cantor-action lifting principles to obtain a continuous profinite shift height. Finite-width lamp slabs then satisfy the compatible-metric theorem of Conley–Jackson–Marks–Seward–Tucker-Drob. The full preimage of the embedded integers is treated by a separate finite Borel exhaustion. The auxiliary Borel theorem requires the height factor and allows stabilizers.

The idea in plain English

A group action describes how transformations move points. Two points lie in the same orbit when one can be moved to the other. Hyperfiniteness means that this orbit relation can be built from an increasing sequence of Borel equivalence relations whose classes are finite.

The group here is the restricted wreath product ℤ ≀ ℤ: an integer shift acts on finitely supported configurations of integer-valued lamps. The paper shows that a generic continuous action on Cantor space has a hyperfinite orbit relation on the whole space. Generic refers to Baire category in the space of all these actions, rather than a probability estimate.

The proof constructs compatible height coordinates modulo successively larger factorials. It cuts the action into finite-width slabs and applies established results on Borel asymptotic dimension. A separate construction handles every point whose profinite height is an ordinary integer.

The precise result

Let W=(⊕ᵢ∈ℤ ℤ)⋊ℤ be the countable discrete restricted wreath product. There is a dense Gδ subset of the full Polish space Act(W,2^ℕ) of continuous actions, with the subspace topology from Homeo(2^ℕ)^W and compact-open homeomorphism topologies, such that every action in this subset has a hyperfinite orbit equivalence relation on the entire Cantor space 2^ℕ.

More generally, suppose W acts Borelly on a standard Borel space X and admits an actual Borel map p:X→Ẑ satisfying p(gx)=p(x)+χ(g), where χ:W→ℤ is the shift homomorphism and Ẑ=lim ℤ/n!ℤ is the group of profinite integers. Then its orbit equivalence relation is hyperfinite on all of X. The theorem permits stabilizers and does not require freeness, minimality, an invariant probability measure or surjectivity of p.

The main theorem answers Question 3 in Iyer’s contribution, joint with Shinko, to Oberwolfach Report 2/2025. This is the historically posed whole-space generic-action question. The auxiliary Borel result requires the height factor; it does not establish hyperfiniteness for every Borel W-action.

Why the result holds

The paper specializes established Cantor-action lifting principles. A direct construction gives compatible height maps modulo successive factorials, using countably many open dense extension conditions in the full action space. Their inverse limit is a continuous equivariant height map; in this compact Cantor setting it is onto.

At each fixed slab width, the lamp directions form a finite-rank free abelian group. Deep results of Conley, Jackson, Marks, Seward and Tucker-Drob give finite Borel asymptotic dimension for its action, including actions with stabilizers. The dimension may grow with the width, and slab components may be infinite. Divisible widths give nested slab graphs with decreasing distances, supplying the exact compatible-metric hypothesis of their Theorem 7.3.

On the full preimage of the embedded integers, which may be uncountable, the proof normalizes to height zero. It combines a finite Borel exhaustion for the lamp subgroup with increasing bounded intervals of integer heights. The height cutoff is essential to keep the pulled-back classes finite. The regular and exceptional pieces are invariant and their finite exhaustions can then be combined.

Scope and limitations

Genericity concerns a dense Gδ subset of the full space of continuous Cantor actions. Hyperfiniteness holds on every point of the phase space for those actions. The statement does not merely discard a null or meager subset of points, and it does not identify the entire set of hyperfinite actions as Gδ.

The general Borel theorem assumes an actual equivariant profinite height map. Stabilizers and nonsurjective maps are allowed. The onto height obtained for generic continuous Cantor actions is a consequence of that compact setting, rather than an added assumption in the Borel theorem. The paper does not treat all Borel W-actions without a height factor.

Finite Borel asymptotic dimension of each slab does not mean that its orbit components are finite. Dimension bounds need not be uniform across widths. The proof uses a specific compatible-metric theorem, rather than a general theorem about increasing unions of hyperfinite relations.

The proof relies on deep prior Borel theorems. Finite diagnostics in the supporting materials check local group and slab behavior; they do not prove the Baire-category or infinite Borel assertions. No comeager conjugacy class, uniform Borel choice of heights or exhaustions across actions, or clopen finite exhaustion is claimed.

No historical firstness, continued worldwide openness, independent human peer review or formal proof-assistant certification is claimed.

Status and AI use

OpenAI tools were used extensively in developing and checking the argument, drafting the note, writing diagnostics and conducting adversarial AI reviews. These checks do not constitute independent human or journal peer review.

The manuscript and Zenodo metadata give 13 October 2026 and version 1.0.0. The record was created on 14 October 2026 UTC. These fields do not establish first priority.

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Cite this paper

Alec Kriebel. (2026). Generic Cantor actions of Z wr Z are hyperfinite (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23356964

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Background and prior work

• Iyer, joint with Shinko — Generic actions of countable groups: Printed pages 114–115 distinguish the earlier measure-hyperfinite result from the whole-space question; Question 3 asks about ℤ ≀ ℤ.
• Conley, Jackson, Marks, Seward and Tucker-Drob — Borel asymptotic dimension and hyperfinite equivalence relations: The author manuscript gives the deep imported results: Corollary 5.5 for each fixed finite-rank abelian action, Theorem 7.3 for compatible metrics, and Corollary 7.5 for the lamp group. The journal article appeared in Duke Mathematical Journal 172 (2023), 3175–3226.
• Iyer and Shinko — Asymptotic dimension and hyperfiniteness of generic Cantor actions: The prior generic hyperfiniteness theorem assumes locally finite asymptotic dimension. The pinned manuscript is version 1, September 2024.
• Akin, Glasner and Weiss — A homeomorphism of the Cantor set: The linked 2006 author preprint gives the conditional-lifting predecessor in Theorem 3.3 and equation (3.7). The later 2008 journal paper is titled “Generically there is but one self homeomorphism of the Cantor set”; this link points to the 2006 preprint.
• Doucha — Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups: Proposition 2.12(3) supplies the all-group extension-perturbation background.
• Doucha, Melleray and Tsankov — Dense and comeager conjugacy classes in zero-dimensional dynamics: Proposition 3.2, Definition A.3 and Proposition A.7 provide the finite-extension and existential-closure background. The link selects version 2 (2026); the earlier version 1 appeared in 2025.