Pith. sign in

REVIEW 2 major objections 6 minor 28 references

Hyperfiniteness of the boundary action of virtually special groups

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every virtually special cubical action has a hyperfinite orbit equivalence relation on the Roller boundary.

desk verdict New, correct-in-outline proof that virtually special cubical actions have hyperfinite Roller boundary orbit equivalence relations, with one omitted boundary-extension lemma that should be added before publication. read the letter →

arxiv 2509.04613 v2 pith:TCJ2XAV2 submitted 2025-09-04 math.GR math.GTmath.LO

Koichi Oyakawa This is my paper · ORCID
classification math.GRmath.GTmath.LO MSC 20F6503E1537A20
keywords hyperfiniteequivalencerelationRollerboundaryCAT(0)cubecomplexvirtuallyspecialgroupsright-angledArtinorbitcubulatedhyperbolicdescriptivesettheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that for any countable group G acting virtually specially on a CAT(0) cube complex X, the orbit equivalence relation induced by G on the Roller boundary of X is hyperfinite: it can be written as an increasing union of finite Borel equivalence relations. This is the first hyperfiniteness result for Roller boundaries of non-hyperbolic cubical actions, and it generalizes the Huang–Sabok–Shinko theorem that cubulated hyperbolic groups act hyperfinitely on their Gromov boundaries. The proof proceeds by encoding Roller boundary points of a right-angled Artin group's cube complex as ordered sequences of labeled hyperplanes, reducing the orbit relation to a tail equivalence relation, and then transferring hyperfiniteness from RAAG actions to virtually special actions through a convex embedding into a RAAG cube complex. If correct, the theorem gives a uniform answer for a large class of nonpositively curved group actions: their natural boundary orbit relations are as simple as Borel equivalence relations can be in this hierarchy.

What carries the argument

The machine is the boundary coding for right-angled Artin groups. Let $\Gamma$ be a finite graph, $A(\Gamma)$ the right-angled Artin group, $X(\Gamma)$ its standard CAT(0) cube complex, and $\mathrm{BR}X(\Gamma)$ the Roller boundary. Hyperplanes of $X(\Gamma)$ are partitioned into finitely many label classes $H_i$, one per generator; hyperplanes in one class never cross, so the hyperplanes in $H_i$ separating a fixed vertex $o$ from a boundary point $\xi$ can be listed in a canonical order by distance from $o$. The proof shows that the $A(\Gamma)$-orbit of a Roller boundary point is determined by the tail of this ordered hyperplane sequence. The tail equivalence relation on the countable set of hyperplanes is hyperfinite by [DJK94, Theorem 8.1], giving Proposition 3.12. The convex projections $\psi_A^B$ between convex sets and the stabilization of cosets $C_n \subseteq C_{n-1} \subseteq \cdots$ (finite graph forces eventual constancy) provide the invariant used to prove smoothness of the orbit relation on hyperplane sequences. For the general theorem, a local isometry $X/H \to R(\Gamma)$ lifts to a convex embedding $X \to X(\Gamma)$, and the induced injection $p_\psi \colon \mathrm{BR}X \to \mathrm{BR}X(\Gamma)$ carries the RAAG hyperfiniteness back.

What would settle it

Find a convex embedding $X \hookrightarrow Y$ of CAT(0) cube complexes such that two distinct points of $\mathrm{BR}X$ are identified in $\mathrm{BR}Y$, or such that the induced boundary map fails to be equivariant with respect to the relevant subgroups; if such an example exists, equation (13) and the reduction to RAAG boundary actions collapse. If no such example exists, a proof of boundary injectivity for convex embeddings would close the gap.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: if X is a CAT(0) cube complex and G is countable and acts virtually specially on X, then the orbit equivalence relation $E^{{BRX}}$_G on the Roller boundary is hyperfinite. The stronger Theorem 4.1 isolates the mechanism: a finite-index subgroup H acting freely with special compact quotient X/H suffices, because Proposition 2.14 transfers hyperfiniteness across finite-index supergroups. The RAAG case (Proposition 3.12) is the engine: for a finite graph Γ, the A(Γ)-action on the Roller boundary of X(Γ) is hyperfinite. Section 5 then verifies the advertised generalization by exhibiting a continuous, surjective, finite-to-one, equivariant map from the Roller boundary of a uniformly locally finite hyperbolic CAT(0) cube complex to its Gromov boundary, so hyperfiniteness of the former is equivalent to hyperfiniteness of the latter.

Load-bearing premise

The load-bearing premise is that a convex copy of one cube complex inside another sends distinct Roller-boundary points to distinct boundary points and respects the group action; the cited lemma establishes only the embedding at the level of complexes, not this boundary extension.

Editorial extensions

If this is right

  • Every cocompactly cubulated hyperbolic group acts hyperfinitely on the Roller boundary of its cube complex; by the Section 5 equivalence, this recovers the Huang–Sabok–Shinko theorem on Gromov boundaries.
  • For any virtually special action, the induced action on the Gromov boundary of the contact graph is hyperfinite (Corollary 4.3).
  • For every right-angled Artin group $A(\Gamma)$, the Roller-boundary action on $X(\Gamma)$ is hyperfinite, and so is the action on the Gromov boundary of its extension graph (Proposition 3.12 and Corollary 4.4).
  • Since hyperfinite implies measure-hyperfinite, every boundary action covered by the theorem is measure-hyperfinite with respect to every Borel probability measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The transition from the convex embedding of complexes to the injection $p_\psi$ of Roller boundaries in Section 4 is asserted rather than proved; a testable question is whether every convex subcomplex inclusion of CAT(0) cube complexes induces an injective Roller-boundary map, and if so, whether the same argument extends to arbitrary convex subcomplex inclusions.
  • The RAAG coding depends only on finitely many pairwise non-crossing label classes and on eventual stabilization of convex-projection cosets, so the same tail-equivalence argument should apply to any cubical action whose hyperplane system admits such a finite labeling, a class potentially wider than virtually special groups.
  • Proposition 5.5 sets up a transfer principle: for hyperbolic CAT(0) cube complexes, hyperfiniteness of the Roller-boundary action is equivalent to hyperfiniteness of the Gromov-boundary action. One could test whether a similar finite-to-one equivariant transfer holds for other pairs of boundary constructions, such as the regular boundary and the full Roller boundary.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proves that if a countable group G acts virtually specially on a CAT(0) cube complex X, then the orbit equivalence relation induced by G on the Roller boundary BRX is hyperfinite (Theorem 1.2). This generalizes the hyperfiniteness theorem of Huang–Sabok–Shinko for boundary actions of cubulated hyperbolic groups. The proof proceeds in three parts. Section 3 treats the case of right-angled Artin groups from scratch: Lemma 3.9 constructs an explicit invariant f for the action of A(Γ) on the space H^N of hyperplane sequences and shows that f separates orbits exactly, so the orbit relation is smooth; Corollary 3.10 and Proposition 3.12 then derive hyperfiniteness of the action on the Roller boundary BRX(Γ) using tail equivalence relations. Section 4 reduces the general case to the RAAG case: a finite-index subgroup H of G acts freely with special quotient Y, the Haglund–Wise theorem gives a local isometry ψ : Y → R(Γ), its lift rψ : X → X(Γ) is a convex embedding by Lemma 2.26, and the paper asserts that rψ induces an injection pψ : BRX → BRX(Γ) satisfying equation (13), which identifies the H-orbit relation on BRX with the ψ_*(π1(Y,y))-orbit relation on pψ(BRX). Section 5 shows that the main theorem indeed generalizes [HSS20], via a finite-to-one surjective equivariant Borel map from the Roller boundary to the Gromov boundary of a hyperbolic CAT(0) cube complex (Lemmas 5.1–5.4 and Proposition 5.5).

Significance. If the proof is completed as indicated, this is a substantial and natural generalization of [HSS20], extending hyperfiniteness of boundary actions from cubulated hyperbolic groups to all countable groups admitting a virtually special cubical action, and it bears directly on Question 1.1 (measure-hyperfinite versus hyperfinite). The RAAG case is the core contribution and is genuinely from scratch: Lemma 3.9 gives an explicit, parameter-free smoothness witness for the action on the hyperplane sequences, with the disk-diagram lemmas 3.2–3.5 supplying the geometry, and there is no fitting, hidden assumption, or use of the target theorem as an input. The applications in Corollaries 4.2–4.4 (hyperbolic groups, contact graphs, extension graphs) are attractive and clearly derived. The descriptive set theory is standard and carefully cited (Dougherty–Jackson–Kechris, Jackson–Kechris–Louveau), and the paper is honest about which steps are routine.

major comments (2)
  1. [Section 4 (after Lemma 2.26; Eq. (13))] The reduction to the RAAG case hinges on the assertion 'rψ induces the injection pψ : BRX → BRX(Γ)', followed by equation (13). Lemma 2.26, as cited, establishes only that the lift rψ is an embedding of X as a convex subcomplex of X(Γ); it says nothing about Roller boundaries. To make the reduction work, the paper needs to prove: (a) existence of a canonical extension of Roller boundary points from the convex subcomplex rψ(X) to X(Γ) (an ambient hyperplane either crosses rψ(X), in which case it restricts to a hyperplane of X, or lies entirely on one side of rψ(X)); (b) injectivity (distinct points of BRX are separated by a hyperplane of X, whose ambient counterpart separates their images); (c) equivariance with respect to the π1(Y,y)-action and the ψ_*(π1(Y,y))-action, using Remark 2.27; and (d) continuity, hence Borelness, of pψ and Borelness of the image pψ(BRX), for example via Lusin–Souslin. None of this appears in the manuscript, and equation (13), which identifies the orbit relation of H on BRX with the pulled-back orbit relation of ψ_*(π1(Y,y)) on pψ(BRX), is asserted without derivation. The statement is very likely correct, but as written a careful reader cannot verify the central reduction; please add this as a lemma with a full proof.
  2. [Section 3 (Lemma 3.9 and Proposition 3.12)] Smoothness of E^{H^N}_{A(Γ)} requires the invariant map to be Borel, but the Borelness of the map f constructed in Lemma 3.9 is never stated or proved; the construction is an inductive procedure (the sets A_n, B_n, C_n, D_n, the stabilization C⃗_h, and the words s_n, t_n) whose measurability is not self-evident and should be justified. Likewise, in Proposition 3.12 the Borelness of the partition sets BRX(Γ)_I and of the map f : BRX(Γ)_I → (H^N)^I is only asserted with 'it is not difficult to see'. These measurability facts are load-bearing: they are what allow Lemma 3.1 and Corollary 3.10 to conclude hyperfiniteness. The arguments are likely short (each step is definable from countable data: the A(Γ)-orbit of a hyperplane and the ordering of H_i(o,ξ)), but they should be written down explicitly.
minor comments (6)
  1. [Theorem 4.1 (preamble)] The sentence 'any finite dimensional CAT(0) cube complex with countably many hyperplanes is countable' is asserted without proof. A short argument suffices: fixing a base vertex o, every vertex v of the 1-skeleton is connected to o by a finite path, so v is separated from o by only finitely many hyperplanes; hence v is determined by a finite subset of the countable set H(X), and X(0) is countable. Please include this argument.
  2. [Lemma 5.1 (proof)] The step 'by applying Theorem 2.15 and Theorem 2.16 to the set {(x,f(x)) : x ∈ X}, there exists a Borel subset A ⊂ X such that f|A : A → Y is Borel isomorphic' is too compressed. The natural argument is to Borel-uniformize the fibers f^{-1}(y) with Lusin–Novikov, obtaining a Borel section s : Y → X, and then note that A = s(Y) is Borel by Lusin–Souslin since s is injective and Borel. Please spell this out and clarify the role of Arsenin–Kunugui.
  3. [Lemma 5.3(2), (iii) ⇒ (i)] The step 'By f(x) = f(y), we have d_X(p_n,q_n) ≤ δ' is asserted without justification. It follows from Proposition 2.30(1) by applying condition (1) to the geodesic triangle with vertices o, p_N, q_M for N, M large enough that n ≤ (p_N,q_M)^S_o; please include this line, since the displayed inequality is otherwise unmotivated.
  4. [Proposition 3.12] The displayed equivalence preceding 'we can see that BRX(Γ)_I is Borel' appears garbled in the manuscript (the second half of the display does not parse). Please rewrite this line cleanly, and also justify explicitly that every ξ ∈ BRX(Γ) belongs to some BRX(Γ)_I with I nonempty (i.e., that H(o,ξ) is infinite for boundary points, so some H_i(o,ξ) is infinite).
  5. [Lemma 5.2(4)] In the proof of Lemma 5.2(4), the inequality |F| + 2 ≤ |Lk_G(p_N)| assumes that the points supplied by Lemma 5.2(3) for the elements of F are distinct from p_{N-1} and p_{N+1} and from each other. This should be said explicitly; distinctness follows because the auxiliary edges are dual to distinct hyperplanes and hyperplanes in a CAT(0) cube complex do not self-osculate.
  6. [Remark 3.13] Remark 3.13 sketches an alternative route through the quotient BRX(Γ)/G and cites [Oya25, Theorem 1.1]. Since the remark is not used in the proof of Proposition 3.12, either delete it or state explicitly that it is optional; the main text should be self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the RAAG boundary action is proved from scratch and the general case reduces to it via Haglund-Wise; the only caveat is an unproved boundary-extension assertion, which is a proof gap, not a circularity.

full rationale

The derivation chain is self-contained at the level where circularity would matter. Section 3 proves hyperfiniteness for the Roller boundary action of right-angled Artin groups directly: Lemma 3.9 establishes smoothness of the action on the hyperplane space H^N, Corollary 3.10 derives hyperfiniteness of the auxiliary relations F_n from tail equivalence, and Proposition 3.12 assembles these into hyperfiniteness of E^{BR X(Γ)}_{A(Γ)}. Crucially, this proof does not invoke Theorem 1.2, the virtually special case, or the HSS20 theorem; it works from the definitions and standard facts about CAT(0) cube complexes and Borel equivalence relations. Section 4 then reduces the general virtually special case to this RAAG case using Theorem 2.24 from Haglund-Wise, which is an external structural result, together with the standard covering-theoretic identifications (12) and the convex embedding Lemma 2.26. The paper asserts without proof that the convex embedding rψ extends to an injection pψ : BR X -> BR X(Γ) and that equation (13) identifies the orbit relations; this is a genuine missing justification, and the reader's skeptical note correctly identifies it as load-bearing. However, it is not circular: the boundary extension is not defined in terms of hyperfiniteness, and the proof does not assume the target conclusion. A gap in a reduction is different from a reduction that is equivalent to its input by construction. Self-citations appear in supporting roles: [Oya24] is used for a geodesic-merging fact in Lemma 5.3, and [Oya25] is used for an example and for an alternative route in Remark 3.13; neither imports Theorem 1.2 or the RAAG result as an unexamined premise. Section 5 derives the HSS20 theorem as a forward consequence of Theorem 1.2 and Proposition 5.5, which is the opposite of circularity. There are no fitted parameters presented as predictions, no ansatz smuggled in via citation, no uniqueness theorem imported from the author's prior work, and no renaming of a known result as a new derivation. Accordingly, the circularity score is 0; the appropriate criticism, if any, is about a missing lemma, not about circular reasoning.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

All free parameters are absent: the proof introduces no fitted constants. The main theorem rests on standard descriptive set theory, standard CAT(0) cube complex structural results, and one unproved geometric assertion about Roller boundaries. No new particles, forces, dimensions, or entities are postulated; the map between Roller and Gromov boundaries is a construction inside known objects.

assumptions (9)
  • standard math Tail equivalence relation on a countable alphabet is hyperfinite; hypersmooth on arbitrary standard Borel spaces (DJK94, Theorem 8.1).
    Used in Corollary 3.10 to show the tail relation on hyperplane sequences is hyperfinite, which is the base of the RAAG argument.
  • standard math Finite index extensions, restrictions, and countable-to-one Borel reductions preserve hyperfiniteness for countable Borel equivalence relations (Proposition 2.14, Lemma 3.1, DJK94).
    Used to pass from G to a finite-index subgroup H and to reduce the general case to the RAAG case.
  • standard math Lusin-Novikov and Arsenin-Kunugui uniformization theorems for countable and Kσ sections.
    Used in Lemma 5.1 to produce a Borel section of the finite-to-one boundary map.
  • domain assumption Median property and disk diagram/corner move machinery for CAT(0) cube complexes (Sageev, Wise).
    Supports Lemma 3.2 and Corollaries 3.4 and 3.5 on convex projections and translations between hyperplane cosets.
  • domain assumption Haglund-Wise Theorem 2.24: a special NPC cube complex with finitely many immersed hyperplanes admits a local isometry into a Salvetti complex.
    This is the bridge from arbitrary virtually special actions to right-angled Artin groups in Section 4.
  • domain assumption A convex embedding of CAT(0) cube complexes extends to an injective map of Roller boundaries and preserves the orbit equivalence relations in the sense of equation (13).
    Asserted without proof in Section 4; this is the weakest visible premise of the main reduction.
  • domain assumption A compact cube complex has finitely many immersed hyperplanes, and a virtually special action yields a finite-dimensional locally finite universal cover with countably many hyperplanes.
    Justifies Theorem 2.24 and the countability of X in Theorem 4.1.
  • domain assumption Genevois Proposition A.2 and Oya24 Lemma 3.27 on the existence and merging of geodesic rays in hyperbolic CAT(0) cube complexes.
    Used in Section 5 to define and control the map from the Roller boundary to the Gromov boundary.
  • domain assumption Uniform local finiteness and hyperbolicity assumptions in Proposition 5.5 make the boundary map f finite-to-one and surjective.
    Needed to transfer hyperfiniteness between Roller and Gromov boundary equivalence relations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hyperfiniteness of the boundary action of virtually special groups." pith.science (2026). https://pith.science/paper/TCJ2XAV2

@misc{pith2026250904613,
  author       = {Pith},
  title        = {Pith review of: Hyperfiniteness of the boundary action of virtually special groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TCJ2XAV2}},
  note         = {Machine review of arXiv:2509.04613}
}
read the original abstract

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of hyperfiniteness of the boundary action of cubulated hyperbolic groups by Huang-Sabok-Shinko.

Figures

Figures reproduced from arXiv: 2509.04613 by the authors.

Figure 1
Figure 1. Lemma 3.2 Lemma 3.2. Let X be a CAT(0) cube complex and A, B Ă Xp0q be convex sets with A X B “ H. Let a0, a1 P A and b0, b1 P B satisfy dXpa0, b0q “ dXpa1, b1q “ dXpA, Bq. Then, we have dXpa0, a1q “ dXpb0, b1q. Moreover, for any geodesic p from a0 to b0 and any geodesic q from a0 to a1 in X, there exist a geodesic r from a1 to b1 in X, a geodesic s from b0 to b1 in X, and a combinatorial map ψ: r0, ns ˆ r0, ms Ñ X,… view at source ↗
Figure 2
Figure 2. The proof of Lemma 3.9 In the proof of Lemma 3.9 below, we essentially use finiteness of Γ to ensure C⃗h ‰ H (see (11)). Lemma 3.9. Define the action ApΓq ñ HN by pg,phnqnPNq ÞÑ pghnqnPN, then its orbit equivalence relation EHN ApΓq is smooth. Proof. Define a map f : HN Ñ pApΓq Nq 2 ˆ V pΓq N as follows. Let ⃗h “ phnqnPN P HN. For each n P N, let vn P V pΓq be the label of the hyperplane hn (see Remark 2.23) and let… view at source ↗
Figure 3
Figure 3. The proof of Lemma 5.2 (3) Lemma 5.2. Let X be a CAT(0) cube complex. The following hold. (1) Let p “ pp0, p1, . . .q and q “ pq0, q1, . . .q be geodesic rays in X with o “ p0 “ q0 P Xp0q . If p and q converge to the same point in BRX, then for any n P N, there exists m P N such that dXpo, pnq ` dXppn, qmq “ dXpo, qmq. (2) For any x, y P BRX, we have dGpx, yq “ |Hpx, yq|. (3) For any x, y P BRX with Hpx, yq “ thu an… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 23 canonical work pages

  1. [1]

    Ian Agol, The virtual H aken conjecture , Doc. Math. 18 (2013), 1045--1087, With an appendix by Agol, Daniel Groves, and Jason Manning. 3104553

    2013

  2. [2]

    Bridson and Andr\' e Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol

    Martin R. Bridson and Andr\' e Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. 1744486

    1999

  3. [3]

    Mladen Bestvina, Michah Sageev, and Karen Vogtmann (eds.), Geometric group theory, IAS/Park City Mathematics Series, vol. 21, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2014, Including lecture notes from the Graduate Summer School held at the Park City Mathematics Institute (PCMI), Park City, UT, July...

    work page 2014

  4. [4]

    Connes, J

    A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431--450. 662736

    work page 1981

  5. [5]

    Dougherty, S

    R. Dougherty, S. Jackson, and A. S. Kechris, The structure of hyperfinite B orel equivalence relations , Trans. Amer. Math. Soc. 341 (1994), no. 1, 193--225

    work page 1994

  6. [6]

    Joshua Frisch, Alexander Kechris, Forte Shinko, and Zoltán Vidnyánszky, Realizations of countable borel equivalence relations, 2025, Preprint, arXiv:2109.12486

    arXiv 2025

  7. [7]

    Talia Fern\'os, Jean L\'ecureux, and Fr\'ed\'eric Math\'eus, Contact graphs, boundaries, and a central limit theorem for CAT (0) cubical complexes , Groups Geom. Dyn. 18 (2024), no. 2, 677--704. 4729822

    work page 2024

  8. [8]

    Anthony Genevois, Contracting isometries of CAT (0) cube complexes and acylindrical hyperbolicity of diagram groups , Algebr. Geom. Topol. 20 (2020), no. 1, 49--134. 4071367

    work page 2020

Show all 28 references
  1. [9]

    Niblo, Complexes and exactness of certain A rtin groups , Algebr

    Erik Guentner and Graham A. Niblo, Complexes and exactness of certain A rtin groups , Algebr. Geom. Topol. 11 (2011), no. 3, 1471--1495. 2821432

    2011

  2. [10]

    Hagen, Weak hyperbolicity of cube complexes and quasi-arboreal groups, J

    Mark F. Hagen, Weak hyperbolicity of cube complexes and quasi-arboreal groups, J. Topol. 7 (2014), no. 2, 385--418. 3217625

    2014

  3. [11]

    Systems 40 (2020), no

    Jingyin Huang, Marcin Sabok, and Forte Shinko, Hyperfiniteness of boundary actions of cubulated hyperbolic groups, Ergodic Theory Dynam. Systems 40 (2020), no. 9, 2453--2466. 4130811

    2020

  4. [12]

    Wise, Special cube complexes, Geom

    Fr\'ed\'eric Haglund and Daniel T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551--1620. 2377497

    2008

  5. [13]

    Jackson, A

    S. Jackson, A. S. Kechris, and A. Louveau, Countable B orel equivalence relations , J. Math. Log. 2 (2002), no. 1, 1--80

    2002

  6. [14]

    Chris Karpinski, Hyperfiniteness of boundary actions of relatively hyperbolic groups, 2022, To appear in Groups Geom. Dyn

    2022

  7. [15]

    Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol

    Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. 1321597

    1995

  8. [16]

    234, Cambridge University Press, Cambridge, 2025

    , The theory of countable B orel equivalence relations , Cambridge Tracts in Mathematics, vol. 234, Cambridge University Press, Cambridge, 2025. 4837611

    2025

  9. [17]

    Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, Forte Shinko, and Pieter Spaas, Hyperfiniteness for group actions on trees, Proc. Amer. Math. Soc. 152 (2024), no. 9, 3657--3664. 4781963

    2024

  10. [18]

    Sang-Hyun Kim and Thomas Koberda, The geometry of the curve graph of a right-angled A rtin group , Internat. J. Algebra Comput. 24 (2014), no. 2, 121--169. 3192368

    2014

  11. [19]

    Dedicata 202 (2019), 27--43

    Timoth\'ee Marquis, On geodesic ray bundles in buildings, Geom. Dedicata 202 (2019), 27--43. 4001806

    2019

  12. [20]

    Timoth\' e e Marquis and Marcin Sabok, Hyperfiniteness of boundary actions of hyperbolic groups, Math. Ann. 377 (2020), no. 3-4, 1129--1153. 4126891

    2020

  13. [21]

    Amos Nevo and Michah Sageev, The P oisson boundary of CAT (0) cube complex groups , Groups Geom. Dyn. 7 (2013), no. 3, 653--695. 3095714

    2013

  14. [22]

    Petr Naryshkin and Andrea Vaccaro, Hyperfiniteness and B orel asymptotic dimension of boundary actions of hyperbolic groups , Math. Ann. 392 (2025), no. 1, 197--208. 4887757

    2025

  15. [23]

    Sigma 12 (2024), Paper No

    Koichi Oyakawa, Hyperfiniteness of boundary actions of acylindrically hyperbolic groups, Forum Math. Sigma 12 (2024), Paper No. e32, 31. 4715159

    2024

  16. [24]

    Koichi Oyakawa, Borel asymptotic dimension of the roller boundary of finite dimensional cat(0) cube complexes, 2025, Preprint, arXiv:2505.10334

    2025 arXiv

  17. [25]

    Sigma 9 (2021), Paper No

    Piotr Przytycki and Marcin Sabok, Unicorn paths and hyperfiniteness for the mapping class group, Forum Math. Sigma 9 (2021), Paper No. e36, 10. 4252215

    2021

  18. [26]

    London Math

    Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585--617. 1347406

    1995

  19. [27]

    Anush Tserunyan, Introduction to descriptive set theory, 2022, Lecture note, https://www.math.mcgill.ca/atserunyan/Teaching_notes/dst_lectures.pdf

    2022

  20. [28]

    Wise, From riches to raags: 3-manifolds, right-angled A rtin groups, and cubical geometry , CBMS Regional Conference Series in Mathematics, vol

    Daniel T. Wise, From riches to raags: 3-manifolds, right-angled A rtin groups, and cubical geometry , CBMS Regional Conference Series in Mathematics, vol. 117, Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, ...

    2012

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.