Pith. sign in

REVIEW 2 major objections 4 minor 71 references

Dynamical comparison for local homeomorphisms

T0 review · 2 major / 4 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 Minimal local homeomorphisms of finite-dimensional compact spaces satisfy dynamical comparison.

desk verdict New for non-injective local homeomorphisms; zero-dimensional proof is solid, but the higher-dimensional theorem rides on an unverified thin-boundary Section 5 that needs a line-by-line referee. read the letter →

arxiv 2608.13000 v1 pith:AKZ5K7HO submitted 2026-08-13 math.OA math.DS

Shirly Geffen , Shanshan Hua , Julian Kranz This is my paper · ORCID
classification math.OAmath.DS MSC 22A2254H2037B0520J0646L3554F45
keywords dynamicalcomparisonDeaconu-RenaultgroupoidslocalhomeomorphismsthinboundarypropertypartialactionspurelyinfiniteC*-algebrasgroupoidhomologyAH-conjecture
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 establishes dynamical comparison for Deaconu–Renault groupoids of minimal surjective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. Dynamical comparison is a regularity property saying that whenever a closed set is smaller than an open set in the eyes of every invariant measure, the closed set can be cut into finitely many pieces and moved disjointly into the open set by the dynamics. In the non-injective case there are no invariant probability measures at all, so comparison says the whole space can be dynamically compressed into any nonempty open set, a purely infinite, paradoxical regime. The authors prove this by embedding partial actions of non-abelian free groups as large subgroupoids and controlling their boundaries with a new thin boundary property derived from finite dimension. If correct, this gives a dynamical proof that the associated C*-algebras are UCT Kirchberg algebras and, in the Cantor case, verifies the AH-conjecture for these groupoids.

What carries the argument

The central technical object is the thin boundary property for étale groupoids: a basis of open sets whose boundaries are $G$-thin, meaning each boundary is dynamically below every nonempty open set. The paper proves this property for minimal second-countable Hausdorff étale groupoids with compact metrizable unit space of finite covering dimension, without freeness assumptions, using a hierarchy of compact sets $\mathcal{C}_m$ defined by collision sets: $K \in \mathcal{C}_m$ if moving two disjoint pieces of $K$ to the same place produces a collision set of rank lower than $m$; Lemma 5.2 converts membership in $\mathcal{C}_m$ into thinness, Lemma 5.5 produces boundaries in general position by dimension-counting, and Theorem 5.8 combines these to produce boundaries in $\mathcal{C}_{d-1}$. This machinery replaces the empty boundaries available in the zero-dimensional case and lets the free-group partial action argument survive in higher dimension.

What would settle it

Find a minimal surjective non-injective local homeomorphism $T$ of a compact metrizable finite-dimensional space and a closed set $A$ and non-empty open set $U$ for which finitely many open bisections with disjoint ranges inside $U$ cannot cover $A$; the paper claims no such pair exists. A direct place to look is a 2-to-1 expanding map of the circle, checking whether a small closed arc is dynamically below a tiny open arc.

Watch

Extended reading notes

Core claim

The paper's central claim is that every Deaconu–Renault groupoid $G_T$ associated with a minimal surjective local homeomorphism $T \colon X \to X$ of a compact metrizable space $X$ with finite Lebesgue covering dimension satisfies dynamical comparison. When $T$ is non-injective, the proof shows the stronger statement that $G_T$ has no invariant probability measures, so dynamical comparison takes the purely infinite form: every closed subset is dynamically below every non-empty open subset. When $T$ is injective, comparison is already known for minimal homeomorphisms, so the new content is the non-injective case. The argument reduces to maps whose fibers have at least two points, embeds a partial action of a non-abelian free group as an open subgroupoid, proves comparison for that partial action using paradoxical towers, and uses the thin boundary property to control the boundary of the embedding. The same comparison result is then used, via known theorems, to recover that $C^*_r(G_T)$ is a UCT Kirchberg algebra and to verify the AH-conjecture for the Cantor-space case.

Load-bearing premise

The load-bearing premise is that in every minimal second-countable étale groupoid with compact metrizable unit space of finite covering dimension, the boundaries of basic open sets are small enough to be dynamically moved into any nonempty open set, without any freeness assumption; the higher-dimensional part of the proof collapses if this Section 5 assertion has a gap.

Editorial extensions

If this is right

  • Every minimal surjective non-injective local homeomorphism of a compact metrizable finite-dimensional space gives a Deaconu–Renault groupoid with no invariant probability measures, so dynamical comparison holds in the strongest purely infinite form: each closed set is dynamically below each nonempty open set.
  • For any such local homeomorphism, the reduced C*-algebra $C^*_r(G_T)$ is a UCT Kirchberg algebra, recovering a known structural theorem by dynamical methods.
  • When the unit space is the Cantor set, the associated groupoid satisfies the AH-conjecture for essentially principal minimal étale groupoids, recovering the AH-conjecture for graph groupoids as a special case.
  • Every minimal second-countable Hausdorff étale groupoid with compact metrizable unit space of finite covering dimension has the thin boundary property.
  • Every minimal action of a countable discrete group on a compact metrizable finite-dimensional space has the small boundary property, with no freeness assumption.

Reading between the lines

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

  • A natural extension, not pursued in the paper, is that the same free-group embedding plus thin-boundary argument should yield dynamical comparison for any minimal second-countable étale groupoid containing a large amenable partial action of a non-elementary hyperbolic group, not just for Deaconu–Renault groupoids.
  • The freeness-free thin-boundary proof suggests that the small-boundary property for minimal actions on finite-dimensional spaces is a general phenomenon, so regularity results built on small boundaries may survive for non-free actions; the paper only explicitly draws the small-boundary conclusion for group actions.
  • In the zero-dimensional case, the proof via partial actions of free groups combined with recent work on groupoid homology opens a combinatorial route to the AH-conjecture for the separated-graph models of local homeomorphisms; the paper leaves a direct graph-based derivation as an open question.
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 / 4 minor

Summary. The paper proves dynamical comparison for Deaconu–Renault groupoids of minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension (Theorem A, Theorem 4.7). The strategy is to reduce the general case to local homeomorphisms with fibers of size at least two (Section 3), embed suitable partial actions of free groups with large domains (Section 4), and control the boundary of the embedded partial action using a new groupoid version of the thin boundary property (Theorem D, Corollary 5.9). Consequences include a dynamical proof that the associated C*-algebras are UCT Kirchberg algebras, verification of Matui's AH-conjecture for zero-dimensional Deaconu–Renault groupoids via Li's theorem, and a freeness-free small boundary property for minimal group actions (Corollaries B, C, E).

Significance. If correct, this is a substantial result: it places a large class of purely infinite étale groupoids in the dynamical comparison regime, recovers and extends known C*-algebraic results by Carlsen–Thomsen through purely dynamical methods, and identifies a general freeness-free small-boundary phenomenon. The paper's strengths include a clean reduction in Section 3, a concrete use of paradoxical towers from [GGKN23], and an explicit and honest disclosure of AI assistance. The main theorem, however, is conditional on the intricate Section 5 proof of Theorem D; as written, two technical points in that proof are not fully justified. These points are load-bearing for the higher-dimensional case, so the paper cannot be certified in its present form, although the issues appear repairable.

major comments (2)
  1. [Section 5, Lemma 5.2] The proof defines K_P := K ∩ closure(P) and later applies the hypothesis K ∈ C_m to the compact sets K_P, which requires K_P ⊂ s(B_P). However, the refinement obtained from [BK04, Lemma 3.2] is only stated to satisfy P ⊂ s(B_P), not closure(P) ⊂ s(B_P). Since s(B_P) is open, closure(P) can stick out of s(B_P), and then K_P is not contained in the source of the bisection B_P. Please add the missing argument that the cover can be chosen with closure(P) ⊂ s(B_P) for every P, or adjust the construction accordingly. Without this, the C_m collision hypothesis cannot be applied to K_P.
  2. [Section 5, Theorem 5.8] At the induction step, Lemma 5.5 is invoked 'with respect to the inclusion O_k ⊂ O_k ∪ N_k', but Lemma 5.5 requires a closed set A. The natural intended set is A = closure(O_k), and this is admissible only if closure(O_k) ⊂ O_k ∪ N_k; that condition follows from the previously recorded ∂O_k ⊂ N_k, but it is not stated. The same issue recurs in the later closure estimates. Please rewrite this induction step explicitly with A = closure(O_k) and verify the boundary containment at each stage, so that the use of Lemma 5.5 is formally valid.
minor comments (4)
  1. [Section 4, Theorem 4.4] In the proof of Theorem 4.4, the statement 'we have D_{h_i} = D_{a_1} = D_{a_2} = X' is not justified and appears inconsistent with the definition of the partial action by α_{a_i} = T|_{A_i}; for a positive word h_i of length at least two, the domain D_{h_i} is typically a proper subset of X. Since the zero-dimensional case can be deduced from the general theorem, please correct this step or remove the redundant proof.
  2. [Section 3, Lemma 3.4] In the displayed chain proving G(0) ≺ U, the summation indices are misprinted: the middle unions over i should run over j = 0, ..., n_i, not over i. The intended formula is clear but should be corrected.
  3. [Section 5, Theorem 5.8] The proof should explicitly record at the start that N_1 is chosen with N_1 ⊂ O, and that the inclusion N_{k+1} ⊂ N_k is maintained after refining N_{k+1}. These facts are used later in deriving \(\overline{W} \subset O\) and ∂W ⊂ N_k, but they are only implicit in the current text.
  4. [References] The main argument depends essentially on the unpublished preprints [Ste26] and [ES19]. Please confirm that the quoted theorems are publicly available in their final form, and consider stating the precise results used from [Ste26, Theorem 2.2] and [ES19, Theorem 9.6, Corollary 9.7] in the text or an appendix.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is derived from an internally proved thin-boundary theorem plus external one-way citations; the only self-citations are non-load-bearing.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Theorem 4.7 reduces via Corollary 3.5 to the large-fibers case and then applies Theorem 4.5, whose hypotheses are verified using the thin boundary property supplied by Theorem 2.4. Theorem 2.4 is not imported from prior work: it is proved in Section 5 from finite covering dimension through the new C_m hierarchy, Lemma 5.2, Lemma 5.5, Theorem 5.8, and Corollary 5.9. The paradoxical-towers tool is attributed to the authors' earlier paper [GGKN23], but the specific lemma used (Lemma 4.2) is stated and proved in full here, so the self-citation is not load-bearing. The realizations of Deaconu-Renault groupoids by partial actions are cited from [Ste26] and [ES19]/[dCK25], which are external to this author group and are used one-way. Li's theorem [Li25] and the small-boundary techniques of [Sza15, KS20, Lin95, Buc13] are likewise external inputs, not re-statements of the conclusion. No parameter is fitted and no quantity called a prediction is defined in terms of the target result. The definition of G-thin (Definition 2.3) uses the independently defined relation A ≺ U from Definition 2.2, but this is a definitional tool, not an assumption of the theorem; nothing in the proof of Theorem 2.4 assumes dynamical comparison for G_T. The AI statement about ChatGPT-assisted drafts of Lemma 5.2 and Theorem 5.8 is a provenance disclosure, not evidence of circularity; any concerns about the correctness or completeness of those proofs are mathematical-gap issues, which are outside the circularity criterion and would not raise the circularity score. Accordingly, no circular step can be exhibited, and the honest finding is score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities appear. The paper's contribution is a proof from existing definitions plus external results; the external results are listed as axioms. The main internal technical premise is the thin boundary property, whose proof is in Section 5 and which is treated as a theorem rather than an assumption.

assumptions (6)
  • domain assumption Deaconu-Renault groupoids of minimal surjective local homeomorphisms are amenable and topologically principal, per SW16 Lemma 3.5 and ABS24 Lemma 7.5.
    Used in Corollary C to conclude the reduced C*-algebra is nuclear, simple, purely infinite, and UCT; these are external results.
  • domain assumption De Castro-Steinberg realization: a Deaconu-Renault groupoid with totally disconnected unit space is isomorphic to the transformation groupoid of a semi-saturated partial action of a free group, per Ste26 Theorem 2.2.
    This is the basis of the zero-dimensional proof in Theorem 4.4 and of Corollary B; the cited paper is a preprint whose proof is not reproduced here.
  • domain assumption Li's theorem relating dynamical comparison to algebraic K-theory spectra and topological full group homology implies Matui's AH conjecture for groupoids with comparison, per Li25.
    Used to derive Corollary B; this is an external published result.
  • domain assumption Partial action groupoid amenability and isomorphism results of Exel-Steinberg, per ES19 Theorem 9.6 and Corollary 9.7, and de Castro-Kang, per dCK25 Theorem 4.1.
    Used to identify the embedded F_d partial action with the Deaconu-Renault subgroupoid and to ensure amenability in Theorems 4.4 and 4.7.
  • standard math Finite-dimensional general-position tools, including BK04 Lemma 3.2 on color refinement with disjoint closures and Eng78/Sza15 Lemma 3.3 on F-sigma sets that lower dimension.
    Used in Section 5 to construct thin boundaries; these are standard dimension-theoretic facts.
  • standard math Gromov ping-pong and north-south dynamics for hyperbolic groups, per Gro87 and AD19 Proposition 4.1.
    Used in Lemma 4.2 to build paradoxical towers for free subsemigroups.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical comparison for local homeomorphisms." pith.science (2026). https://pith.science/paper/AKZ5K7HO

@misc{pith2026260813000,
  author       = {Pith},
  title        = {Pith review of: Dynamical comparison for local homeomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKZ5K7HO}},
  note         = {Machine review of arXiv:2608.13000}
}
abstract

We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C^*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 26 canonical work pages

  1. [1]

    Amenability and comparison for \'etale groupoids with polynomial growth

    Are Austad and Christian B \"o nicke. Amenability and comparison for \'etale groupoids with polynomial growth, 2026. Preprint. https://arxiv.org/abs/2605.16013 arXiv:2605.16013

    work page Pith review arXiv 2026

  2. [2]

    Strict comparison for C^* -algebras arising from almost finite groupoids

    Pere Ara, Christian B\" o nicke, Joan Bosa, and Kang Li. Strict comparison for C^* -algebras arising from almost finite groupoids. Banach J. Math. Anal. , 14(4):1692--1710, 2020. https://doi.org/10.1007/s43037-020-00079-6 doi:10.1007/s43037-020-00079-6

    doi:10.1007/s43037-020-00079-6 2020

  3. [3]

    The type semigroup, comparison, and almost finiteness for ample groupoids

    Pere Ara, Christian B\" o nicke, Joan Bosa, and Kang Li. The type semigroup, comparison, and almost finiteness for ample groupoids. Ergodic Theory Dynam. Systems , 43(2):361--400, 2023. https://doi.org/10.1017/etds.2021.115 doi:10.1017/etds.2021.115

    doi:10.1017/etds.2021.115 2023

  4. [4]

    Simplicity of twisted C^* -algebras of D eaconu- R enault groupoids

    Becky Armstrong, Nathan Brownlowe, and Aidan Sims. Simplicity of twisted C^* -algebras of D eaconu- R enault groupoids. J. Noncommut. Geom. , 18(1):265--312, 2024. https://doi.org/10.4171/jncg/527 doi:10.4171/jncg/527

    work page doi:10.4171/jncg/527 2024

  5. [5]

    A correspondence between surjective local homeomorphisms and a family of separated graphs

    Pere Ara and Joan Claramunt. A correspondence between surjective local homeomorphisms and a family of separated graphs. Discrete Contin. Dyn. Syst. , 44(5):1178--1266, 2024. https://doi.org/10.3934/dcds.2023143 doi:10.3934/dcds.2023143

    work page doi:10.3934/dcds.2023143 2024

  6. [6]

    Purely infinite C^* -algebras arising from dynamical systems

    Claire Anantharaman-Delaroche. Purely infinite C^* -algebras arising from dynamical systems. Bull. Soc. Math. France , 125(2):199--225, 1997. https://doi.org/10.24033/bsmf.2304 doi:10.24033/bsmf.2304

    work page doi:10.24033/bsmf.2304 1997

  7. [7]

    Abbott and Fran c ois Dahmani

    Carolyn R. Abbott and Fran c ois Dahmani. Property P_ naive for acylindrically hyperbolic groups. Math. Z. , 291:555--568, 2019. https://doi.org/10.1007/s00209-018-2094-1 doi:10.1007/s00209-018-2094-1

    doi:10.1007/s00209-018-2094-1 2019

  8. [8]

    Amenable groupoids , volume 36 of Monographies de L'Enseignement Math\'ematique [Monographs of L'Enseignement Math\'ematique]

    Claire Anantharaman-Delaroche and Jean Renault. Amenable groupoids , volume 36 of Monographies de L'Enseignement Math\'ematique [Monographs of L'Enseignement Math\'ematique] . L'Enseignement Math\'ematique, Geneva, 2000. With a foreword by Georges Skandalis and Appendix B by E. Germain

    work page 2000

Show all 71 references
  1. [9]

    Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions

    Pere Ara and Ruy Exel. Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions. Adv. Math. , 252:748--804, 2014. https://doi.org/10.1016/j.aim.2013.11.009 doi:10.1016/j.aim.2013.11.009

    2014 doi

  2. [10]

    Purely infinite simple reduced C^* -algebras of one-relator separated graphs

    Pere Ara. Purely infinite simple reduced C^* -algebras of one-relator separated graphs. J. Math. Anal. Appl. , 393(2):493--508, 2012. https://doi.org/10.1016/j.jmaa.2012.04.014 doi:10.1016/j.jmaa.2012.04.014

    2012 doi

  3. [11]

    Stable rank one in nonnuclear crossed products, 2025

    Jamie Bell, Shirly Geffen, and David Kerr. Stable rank one in nonnuclear crossed products, 2025. Preprint. https://arxiv.org/abs/2511.20132 arXiv:2511.20132

    2025

  4. [12]

    Bridson and Andr\'e Haefliger

    Martin R. Bridson and Andr\'e Haefliger. Metric spaces of non-positive curvature , volume 319 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1999. https://doi.org/10.1007/978-3-662-12494-9 doi:10.10...

    1999 doi

  5. [13]

    Global G limm halving for C^* -bundles

    Etienne Blanchard and Eberhard Kirchberg. Global G limm halving for C^* -bundles. J. Operator Theory , 52(2):385--420, 2004. URL: https://jot.theta.ro/jot/archive/2004-052-002/2004-052-002-011.html

    2004

  6. [14]

    Ideal structure and pure infiniteness of ample groupoid C^* -algebras

    Christian B \"o nicke and Kang Li. Ideal structure and pure infiniteness of ample groupoid C^* -algebras. Ergodic Theory Dynam. Systems , 40(1):34--63, 2020. https://doi.org/10.1017/etds.2018.39 doi:10.1017/etds.2018.39

    2020 doi

  7. [15]

    Topologically free minimal actions without dynamical comparison, 2026

    Paolo Boldrini and Akshara Prasad. Topologically free minimal actions without dynamical comparison, 2026. Preprint. https://arxiv.org/abs/2607.01896 arXiv:2607.01896

    2026 arXiv

  8. [16]

    The C^* -algebras of row-finite graphs

    Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szyma \'n ski. The C^* -algebras of row-finite graphs. New York J. Math. , 6:307--324, 2000. URL: https://nyjm.albany.edu/j/2000/6-14.pdf

    2000

  9. [17]

    Smallness and comparison properties for minimal dynamical systems, 2013

    Julian Buck. Smallness and comparison properties for minimal dynamical systems, 2013. Preprint. https://arxiv.org/abs/1306.6681 arXiv:1306.6681

    2013 arXiv

  10. [18]

    The structure of the C^* -algebra of a locally injective surjection

    Toke Meier Carlsen and Klaus Thomsen. The structure of the C^* -algebra of a locally injective surjection. Ergodic Theory Dynam. Systems , 32(4):1226--1248, 2012. https://doi.org/10.1017/S0143385711000216 doi:10.1017/S0143385711000216

    2012 doi

  11. [19]

    de Castro and Eun Ji Kang

    Gilles G. de Castro and Eun Ji Kang. A categorical interpretation of continuous orbit equivalence for partial dynamical systems. Bull. Lond. Math. Soc. , 57(12):3637--3656, 2025. https://doi.org/10.1112/blms.70207 doi:10.1112/blms.70207

    2025 doi

  12. [20]

    Groupoids associated with endomorphisms

    Valentin Deaconu. Groupoids associated with endomorphisms. Trans. Amer. Math. Soc. , 347(5):1779--1786, 1995. https://doi.org/10.2307/2154972 doi:10.2307/2154972

    1995 doi

  13. [21]

    Symbolic extensions of amenable group actions and the comparison property

    Tomasz Downarowicz and Guohua Zhang. Symbolic extensions of amenable group actions and the comparison property. Mem. Amer. Math. Soc. , 281(1390), 2023. https://doi.org/10.1090/memo/1390 doi:10.1090/memo/1390

    2023 doi

  14. [22]

    Elliott and Zhuang Niu

    George A. Elliott and Zhuang Niu. The C^* -algebra of a minimal homeomorphism of zero mean dimension. Duke Math. J. , 166(18):3569--3594, 2017. https://doi.org/10.1215/00127094-2017-0033 doi:10.1215/00127094-2017-0033

    2017 doi

  15. [23]

    Dimension theory , volume 19 of North-Holland Mathematical Library

    Ryszard Engelking. Dimension theory , volume 19 of North-Holland Mathematical Library . North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN---Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author. URL: https://www.sciencedire...

    1978

  16. [24]

    Subshift semigroups, 2019

    Ruy Exel and Benjamin Steinberg. Subshift semigroups, 2019. Preprint. https://arxiv.org/abs/1908.08315 arXiv:1908.08315

    2019 arXiv

  17. [25]

    Circle actions on C^* -algebras, partial automorphisms, and a generalized P imsner- V oiculescu exact sequence

    Ruy Exel. Circle actions on C^* -algebras, partial automorphisms, and a generalized P imsner- V oiculescu exact sequence. J. Funct. Anal. , 122(2):361--401, 1994. https://doi.org/10.1006/jfan.1994.1073 doi:10.1006/jfan.1994.1073

    1994

  18. [26]

    Essential freeness, allostery and Z -stability of crossed products, 2024

    Eusebio Gardella, Shirly Geffen, Rafaela Gesing, Grigoris Kopsacheilis, and Petr Naryshkin. Essential freeness, allostery and Z -stability of crossed products, 2024. Preprint. https://arxiv.org/abs/2405.04343 arXiv:2405.04343

    2024 arXiv

  19. [27]

    Classifiability of crossed products by nonamenable groups

    Eusebio Gardella, Shirly Geffen, Julian Kranz, and Petr Naryshkin. Classifiability of crossed products by nonamenable groups. J. Reine Angew. Math. , 2023(797):285--312, 2023. https://doi.org/10.1515/crelle-2023-0012 doi:10.1515/crelle-2023-0012

    2023 doi

  20. [28]

    Hyperbolic groups

    Mikhail Gromov. Hyperbolic groups. In Essays in group theory , volume 8 of Math. Sci. Res. Inst. Publ. , pages 75--263. Springer, New York, 1987. https://doi.org/10.1007/978-1-4613-9586-7_3 doi:10.1007/978-1-4613-9586-7_3

    1987 doi

  21. [29]

    Purely infinite Cuntz--Krieger algebras of directed graphs

    Jeong Hee Hong and Wojciech Szyma \'n ski. Purely infinite Cuntz--Krieger algebras of directed graphs. Bull. Lond. Math. Soc. , 35(5):689--696, 2003. https://doi.org/10.1112/S0024609303002364 doi:10.1112/S0024609303002364

    2003 doi

  22. [30]

    Cantor systems, piecewise translations and simple amenable groups

    Kate Juschenko and Nicolas Monod. Cantor systems, piecewise translations and simple amenable groups. Ann. of Math. (2) , 178(2):775--787, 2013. https://doi.org/10.4007/annals.2013.178.2.7 doi:10.4007/annals.2013.178.2.7

    2013 doi

  23. [31]

    Extensions of amenable groups by recurrent groupoids

    Kate Juschenko, Volodymyr Nekrashevych, and Mikael de la Salle. Extensions of amenable groups by recurrent groupoids. Invent. Math. , 206(3):837--867, 2016. https://doi.org/10.1007/s00222-016-0664-6 doi:10.1007/s00222-016-0664-6

    2016 doi

  24. [32]

    Dimension, comparison, and almost finiteness

    David Kerr. Dimension, comparison, and almost finiteness. J. Eur. Math. Soc. (JEMS) , 22(11):3697--3745, 2020. https://doi.org/10.4171/jems/995 doi:10.4171/jems/995

    2020 doi

  25. [33]

    Bauer simplices and the small boundary property

    David Kerr, Grigoris Kopsacheilis, and Spyridon Petrakos. Bauer simplices and the small boundary property. J. Funct. Anal. , 287(11), 2024. https://doi.org/10.1016/j.jfa.2024.110619 doi:10.1016/j.jfa.2024.110619

    2024

  26. [34]

    Elementary amenability and almost finiteness

    David Kerr and Petr Naryshkin. Elementary amenability and almost finiteness. Compos. Math. , 161(12):3321--3337, 2025. https://doi.org/10.1017/S0010437X26102899 doi:10.1017/S0010437X26102899

    2025 doi

  27. [35]

    Almost finiteness and the small boundary property

    David Kerr and G\' a bor Szab\' o . Almost finiteness and the small boundary property. Comm. Math. Phys. , 374(1):1--31, 2020. https://doi.org/10.1007/s00220-019-03519-z doi:10.1007/s00220-019-03519-z

    2020 doi

  28. [36]

    Ample groupoids, topological full groups, algebraic K -theory spectra and infinite loop spaces

    Xin Li. Ample groupoids, topological full groups, algebraic K -theory spectra and infinite loop spaces. Forum Math. Pi , 13(e9), 2025. https://doi.org/10.1017/fmp.2024.31 doi:10.1017/fmp.2024.31

    2025 doi

  29. [37]

    Lowering topological entropy

    Elon Lindenstrauss. Lowering topological entropy. J. Anal. Math. , 67:231--267, 1995. https://doi.org/10.1007/BF02787792 doi:10.1007/BF02787792

    1995 doi

  30. [38]

    Stable rank of C(X) , 2023

    Chun Guang Li and Zhuang Niu. Stable rank of C(X) , 2023. Preprint. https://arxiv.org/abs/2008.03361 arXiv:2008.03361

    2023 arXiv

  31. [39]

    Mean topological dimension

    Elon Lindenstrauss and Benjamin Weiss. Mean topological dimension. Israel J. Math. , 115:1--24, 2000. https://doi.org/10.1007/BF02810577 doi:10.1007/BF02810577

    2000 doi

  32. [40]

    Purely infinite locally compact H ausdorff \'etale groupoids and their C^* -algebras

    Xin Ma. Purely infinite locally compact H ausdorff \'etale groupoids and their C^* -algebras. Int. Math. Res. Not. IMRN , 2022(11):8420--8471, 2022. https://doi.org/10.1093/imrn/rnaa360 doi:10.1093/imrn/rnaa360

    2022 doi

  33. [41]

    Some remarks on topological full groups of Cantor minimal systems

    Hiroki Matui. Some remarks on topological full groups of Cantor minimal systems. Internat. J. Math. , 17(2):231--251, 2006. https://doi.org/10.1142/S0129167X06003448 doi:10.1142/S0129167X06003448

    2006 doi

  34. [42]

    Homology and topological full groups of \' e tale groupoids on totally disconnected spaces

    Hiroki Matui. Homology and topological full groups of \' e tale groupoids on totally disconnected spaces. Proc. Lond. Math. Soc. (3) , 104(1):27--56, 2012. https://doi.org/10.1112/plms/pdr029 doi:10.1112/plms/pdr029

    2012 doi

  35. [43]

    \'E tale groupoids arising from products of shifts of finite type

    Hiroki Matui. \'E tale groupoids arising from products of shifts of finite type. Adv. Math. , 303:502--548, 2016. https://doi.org/10.1016/j.aim.2016.08.023 doi:10.1016/j.aim.2016.08.023

    2016 doi

  36. [44]

    Almost elementary \'etale groupoids, 2026

    Xin Ma and Jianchao Wu. Almost elementary \'etale groupoids, 2026. Preprint. https://arxiv.org/abs/2011.01182 arXiv:2011.01182

    2026 arXiv

  37. [45]

    Polynomial growth, comparison, and the small boundary property

    Petr Naryshkin. Polynomial growth, comparison, and the small boundary property. Adv. Math. , 406, 2022. https://doi.org/10.1016/j.aim.2022.108519 doi:10.1016/j.aim.2022.108519

    2022

  38. [46]

    Group extensions preserve almost finiteness

    Petr Naryshkin. Group extensions preserve almost finiteness. J. Funct. Anal. , 286(7), 2024. https://doi.org/10.1016/j.jfa.2024.110348 doi:10.1016/j.jfa.2024.110348

    2024

  39. [47]

    URP , comparison, mean dimension, and sharp shift embeddability, 2024

    Petr Naryshkin. URP , comparison, mean dimension, and sharp shift embeddability, 2024. Preprint. https://arxiv.org/abs/2410.01757 arXiv:2410.01757

    2024 arXiv

  40. [48]

    Palindromic subshifts and simple periodic groups of intermediate growth

    Volodymyr Nekrashevych. Palindromic subshifts and simple periodic groups of intermediate growth. Ann. of Math. (2) , 187(3):667--719, 2018. https://doi.org/10.4007/annals.2018.187.3.2 doi:10.4007/annals.2018.187.3.2

    2018 doi

  41. [49]

    Comparison radius and mean topological dimension: R okhlin property, comparison of open sets, and subhomogeneous C^* -algebras

    Zhuang Niu. Comparison radius and mean topological dimension: R okhlin property, comparison of open sets, and subhomogeneous C^* -algebras. J. Anal. Math. , 146(2):595--672, 2022. https://doi.org/10.1007/s11854-022-0205-8 doi:10.1007/s11854-022-0205-8

    2022 doi

  42. [50]

    Matui's AH conjecture for graph groupoids

    Petter Nyland and Eduard Ortega. Matui's AH conjecture for graph groupoids. Doc. Math. , 26:1679--1727, 2021. https://doi.org/10.4171/DM/853 doi:10.4171/DM/853

    2021 doi

  43. [51]

    Almost finiteness and groups of dynamical origin

    Petr Naryshkin and Spyridon Petrakos. Almost finiteness and groups of dynamical origin. Int. Math. Res. Not. IMRN , 2025(3), 2025. https://doi.org/10.1093/imrn/rnaf016 doi:10.1093/imrn/rnaf016

    2025 doi

  44. [52]

    Nuclear dimension, pure infiniteness and real rank for higher-rank graph C^* -algebras, 2026

    David Pask. Nuclear dimension, pure infiniteness and real rank for higher-rank graph C^* -algebras, 2026. Preprint. https://arxiv.org/abs/2607.27691 arXiv:2607.27691

    2026 arXiv

  45. [53]

    Rank-two graphs whose C^* -algebras are direct limits of circle algebras

    David Pask, Iain Raeburn, Mikael R rdam, and Aidan Sims. Rank-two graphs whose C^* -algebras are direct limits of circle algebras. J. Funct. Anal. , 239(1):137--178, 2006. https://doi.org/10.1016/j.jfa.2006.04.003 doi:10.1016/j.jfa.2006.04.003

    2006 doi

  46. [54]

    Real rank and topological dimension of higher-rank graph algebras

    David Pask, Adam Sierakowski, and Aidan Sims. Real rank and topological dimension of higher-rank graph algebras. Indiana Univ. Math. J. , 66(6):2137--2168, 2017. https://doi.org/10.1512/iumj.2017.66.6212 doi:10.1512/iumj.2017.66.6212

    2017 doi

  47. [55]

    Structure theory and stable rank for C^* -algebras of finite higher-rank graphs

    David Pask, Adam Sierakowski, and Aidan Sims. Structure theory and stable rank for C^* -algebras of finite higher-rank graphs. Proc. Edinb. Math. Soc. (2) , 64(4):822--847, 2021. https://doi.org/10.1017/S0013091521000626 doi:10.1017/S0013091521000626

    2021 doi

  48. [56]

    A groupoid approach to C^* -algebras , volume 793 of Lecture Notes in Mathematics

    Jean Renault. A groupoid approach to C^* -algebras , volume 793 of Lecture Notes in Mathematics . Springer, Berlin, 1980. https://doi.org/10.1007/BFb0091072 doi:10.1007/BFb0091072

    1980 doi

  49. [57]

    Purely infinite C^* -algebras arising from crossed products

    Mikael R rdam and Adam Sierakowski. Purely infinite C^* -algebras arising from crossed products. Ergodic Theory Dynam. Systems , 32(1):273--293, 2012. https://doi.org/10.1017/S0143385710000829 doi:10.1017/S0143385710000829

    2012 doi

  50. [58]

    A dichotomy for groupoid C^* -algebras

    Timothy Rainone and Aidan Sims. A dichotomy for groupoid C^* -algebras. Ergodic Theory Dynam. Systems , 40(2):521--563, 2020. https://doi.org/10.1017/etds.2018.52 doi:10.1017/etds.2018.52

    2020 doi

  51. [59]

    Homological stability for automorphism groups

    Oscar Randal-Williams and Nathalie Wahl. Homological stability for automorphism groups. Adv. Math. , 318:534--626, 2017. https://doi.org/10.1016/j.aim.2017.07.022 doi:10.1016/j.aim.2017.07.022

    2017 doi

  52. [60]

    Gauge-invariant ideals in the C^* -algebras of finitely aligned higher-rank graphs

    Aidan Sims. Gauge-invariant ideals in the C^* -algebras of finitely aligned higher-rank graphs. Canad. J. Math. , 58(6):1268--1290, 2006. https://doi.org/10.4153/CJM-2006-045-2 doi:10.4153/CJM-2006-045-2

    2006 doi

  53. [61]

    Partial actions of free groups and groupoid homology, 2026

    Benjamin Steinberg. Partial actions of free groups and groupoid homology, 2026. Preprint. https://arxiv.org/abs/2602.15170 arXiv:2602.15170

    2026

  54. [62]

    Almost finiteness for general \' e tale groupoids and its applications to stable rank of crossed products

    Yuhei Suzuki. Almost finiteness for general \' e tale groupoids and its applications to stable rank of crossed products. Int. Math. Res. Not. IMRN , 2020(19):6007--6041, 2020. https://doi.org/10.1093/imrn/rny187 doi:10.1093/imrn/rny187

    2020 doi

  55. [63]

    Can one always lower topological entropy? Ergodic Theory Dynam

    Michael Shub and Benjamin Weiss. Can one always lower topological entropy? Ergodic Theory Dynam. Systems , 11(3):535--546, 1991. https://doi.org/10.1017/S0143385700006325 doi:10.1017/S0143385700006325

    1991 doi

  56. [64]

    Williams

    Aidan Sims and Dana P. Williams. The primitive ideals of some \' e tale groupoid C^* -algebras. Algebr. Represent. Theory , 19(2):255--276, 2016. https://doi.org/10.1007/s10468-015-9573-4 doi:10.1007/s10468-015-9573-4

    2016 doi

  57. [65]

    The homology of the Higman--Thompson groups

    Markus Szymik and Nathalie Wahl. The homology of the Higman--Thompson groups. Invent. Math. , 216(2):445--518, 2019. https://doi.org/10.1007/s00222-018-0848-z doi:10.1007/s00222-018-0848-z

    2019 doi

  58. [66]

    Rachel Skipper, Stefan Witzel, and Matthew C. B. Zaremsky. Simple groups separated by finiteness properties. Invent. Math. , 215(2):713--740, 2019. https://doi.org/10.1007/s00222-018-0835-8 doi:10.1007/s00222-018-0835-8

    2019 doi

  59. [67]

    The R okhlin dimension of topological Z ^m -actions

    G\' a bor Szab\' o . The R okhlin dimension of topological Z ^m -actions. Proc. Lond. Math. Soc. (3) , 110(3):673--694, 2015. https://doi.org/10.1112/plms/pdu065 doi:10.1112/plms/pdu065

    2015 doi

  60. [68]

    La conjecture de B aum- C onnes pour les feuilletages moyennables

    Jean-Louis Tu. La conjecture de B aum- C onnes pour les feuilletages moyennables. K -Theory , 17(3):215--264, 1999. https://doi.org/10.1023/A:1007744304422 doi:10.1023/A:1007744304422

    1999 doi

  61. [69]

    Toms and Wilhelm Winter

    Andrew S. Toms and Wilhelm Winter. The E lliott conjecture for V illadsen algebras of the first type. J. Funct. Anal. , 256(5):1311--1340, 2009. https://doi.org/10.1016/j.jfa.2008.12.015 doi:10.1016/j.jfa.2008.12.015

    2009 doi

  62. [70]

    Abstract classification theorems for amenable C^* -algebras

    Stuart White. Abstract classification theorems for amenable C^* -algebras. In International Congress of Mathematicians , pages 3314--3338. European Mathematical Society-EMS-Publishing House GmbH, 2023. https://doi.org/10.4171/ICM2022/183 doi:10.4171/ICM2022/183

    2023 doi

  63. [71]

    Structure of nuclear C^* -algebras: from quasidiagonality to classification and back again

    Wilhelm Winter. Structure of nuclear C^* -algebras: from quasidiagonality to classification and back again. In Proceedings of the I nternational C ongress of M athematicians--- R io de J aneiro 2018. V ol. III . I nvited lectures , pages 1801--1823. World Sci. Publ., Hackensac...

    2018 doi

Pith tools

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