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arxiv: 2606.21761 · v1 · pith:CBNFXMLMnew · submitted 2026-06-19 · 🧮 math.KT · math.OA

Discretisation and independent resolutions of ample groupoids

Xin Li , Alistair Miller This is my paper

Pith reviewed 2026-06-26 12:48 UTC · model grok-4.3

classification 🧮 math.KT math.OA MSC 46L8022A22
keywords ample groupoidsgroupoid homologytopological K-theorydiscretisationindependent resolutionsGarside categoriesC*-algebrasStein's groups
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The pith

Discretisation shows that independent ample groupoids are equivalent to discrete groupoids in homology and K-theory.

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

The paper sets out a framework that uses discretisation and independent resolutions to compute the groupoid homology of ample groupoids and the topological K-theory of the reduced C*-algebras they generate. It proves that a subclass called independent groupoids matches discrete groupoids exactly in both homology and K-theory. The authors supply an explicit recipe for building controlled independent resolutions of arbitrary ample groupoids, which turns the study of these invariants into a systematic reduction to the discrete case. Examples drawn from Garside categories and Stein groups illustrate how the method applies in practice.

Core claim

Discretisation shows that a special class of ample groupoids termed independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. The paper introduces the notion of a resolution by independent groupoids and provides a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory.

What carries the argument

independent resolutions: controlled resolutions of an ample groupoid by independent groupoids that are equivalent to discrete groupoids, carrying the homology and K-theory data.

If this is right

  • Homology groups of ample groupoids arising from Garside categories reduce to computations on discrete groupoids.
  • K-theory of the reduced C*-algebras of these groupoids is determined by the corresponding discrete case.
  • The method applies directly to higher-rank graphs, self-similar groups, spherical Artin-Tits groups, and Stein groups.
  • Any ample groupoid of interest admits a controlled independent resolution that yields its homology and K-theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique could be tested on ample groupoids outside the Garside setting to check how far the equivalence extends.
  • If the resolutions also preserve other invariants such as cohomology with local coefficients, they would link groupoid theory to a wider range of algebraic computations.
  • The construction supplies a concrete algorithm that could be implemented to produce explicit chain complexes for previously inaccessible groupoids.

Load-bearing premise

The definitions of independent groupoids and the construction of independent resolutions preserve all structural features needed for the homology and K-theory equivalences to hold.

What would settle it

Direct computation of the homology of a concrete ample groupoid from a Garside category that differs from the homology obtained after passing through its independent resolution.

read the original abstract

We develop a general framework for understanding and computing both the groupoid homology of an ample groupoid and the topological K-theory of its reduced C*-algebra, based on two main ideas: discretisation and independent resolutions. Discretisation shows that a special class of ample groupoids we term independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. We introduce the notion of a resolution by independent groupoids and provide a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory. In order to illustrate our general ideas and methods, we work out several concrete examples and applications. Garside categories provide a wide range of examples, including higher rank graphs, self-similar groups and spherical Artin-Tits groups. We also present an application to the homology of Stein's groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper develops a framework for computing groupoid homology and the topological K-theory of the reduced C*-algebra of an ample groupoid, via two constructions: (i) discretisation, which establishes homological and K-theoretic equivalence between a subclass of 'independent' ample groupoids and discrete groupoids, and (ii) controlled independent resolutions, for which the authors supply a general recipe that reduces the study of an arbitrary ample groupoid to the independent case. The framework is illustrated on Garside categories (including higher-rank graphs, self-similar groups and spherical Artin-Tits groups) and on Stein groups.

Significance. If the claimed preservation of both homology and reduced C*-algebra K-theory under discretisation and controlled resolutions holds, the work would supply a systematic computational tool for a wide family of groupoids that appear in geometric group theory and operator algebras. The explicit recipes and concrete examples constitute a genuine contribution to the literature on groupoid invariants.

major comments (3)
  1. [§3] §3 (Discretisation): The assertion that discretisation induces an isomorphism on the K-theory of the reduced C*-algebra (as opposed to merely on groupoid homology) is load-bearing for the central claim, yet the argument appears to rest on the control condition alone; it is not shown that this condition guarantees the necessary exactness or Morita equivalence properties of the reduced crossed product that would yield a KK-equivalence.
  2. [§4.3] §4.3 (Construction of controlled independent resolutions): The recipe for building a controlled resolution is stated to extend the equivalence to general ample groupoids, but the manuscript does not verify that the resolution maps induce isomorphisms on K-theory of the reduced C*-algebra for the target examples (Garside categories, Stein groups); only the homology side is treated in detail.
  3. [§5] §5 (Applications): In the treatment of spherical Artin-Tits groups and Stein groups, the claimed K-theoretic computations rely on the unverified functoriality of the discretisation map with respect to the reduced C*-algebra; without an explicit check that the control condition preserves the reduced crossed-product structure, the K-theory statements remain conditional.
minor comments (2)
  1. [§4] The notation for the control condition on resolutions is introduced without a numbered definition; a displayed definition would improve readability.
  2. Several citations to earlier work on ample groupoids and Garside categories are given only by author names; full bibliographic details should be supplied in the references.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and for identifying points where the K-theoretic arguments require additional clarification. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [§3] §3 (Discretisation): The assertion that discretisation induces an isomorphism on the K-theory of the reduced C*-algebra (as opposed to merely on groupoid homology) is load-bearing for the central claim, yet the argument appears to rest on the control condition alone; it is not shown that this condition guarantees the necessary exactness or Morita equivalence properties of the reduced crossed product that would yield a KK-equivalence.

    Authors: The control condition is introduced precisely to guarantee that the discretisation functor induces a Morita equivalence between the reduced C*-algebras of the independent ample groupoid and its discrete counterpart. This equivalence is constructed explicitly via a bimodule whose existence follows from the independence and control hypotheses (see the argument following Definition 3.2 and the proof of Theorem 3.5). The exactness of the relevant sequences is inherited from the discrete case, which is already known to be KK-equivalent under these restrictions. We agree that the link could be stated more explicitly and will add a dedicated paragraph in the revised §3 spelling out the Morita equivalence step. revision: yes

  2. Referee: [§4.3] §4.3 (Construction of controlled independent resolutions): The recipe for building a controlled resolution is stated to extend the equivalence to general ample groupoids, but the manuscript does not verify that the resolution maps induce isomorphisms on K-theory of the reduced C*-algebra for the target examples (Garside categories, Stein groups); only the homology side is treated in detail.

    Authors: The controlled resolution is constructed so that each step satisfies the control condition, allowing the K-theoretic equivalence to be obtained by composing the individual KK-equivalences. While the homology computation is written out in full for the examples, the K-theory side follows by the same functoriality. We will insert a short verification subsection after the construction in §4.3 confirming that the resolution maps preserve the reduced crossed-product structure for the Garside and Stein-group cases. revision: yes

  3. Referee: [§5] §5 (Applications): In the treatment of spherical Artin-Tits groups and Stein groups, the claimed K-theoretic computations rely on the unverified functoriality of the discretisation map with respect to the reduced C*-algebra; without an explicit check that the control condition preserves the reduced crossed-product structure, the K-theory statements remain conditional.

    Authors: The functoriality of discretisation with respect to the reduced C*-algebra is established in §3 under the control condition and is invoked in §5 only after verifying that the examples satisfy this condition. Nevertheless, to remove any ambiguity we will add an explicit paragraph in each application subsection of §5 confirming that the control condition holds and therefore the reduced crossed products are KK-equivalent, rendering the K-theory statements unconditional. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and constructions are self-contained

full rationale

The paper presents a novel framework based on newly introduced notions of independent groupoids and controlled independent resolutions. These are defined and then used to establish equivalences for homology and K-theory, without any visible reduction of outputs to fitted parameters, self-definitional loops, or load-bearing self-citations in the abstract or described claims. The derivation chain relies on the internal properties of the new constructions rather than renaming known results or smuggling ansatzes via prior author work. This is the expected outcome for a theoretical paper introducing a systematic method without empirical fitting or circular self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper introduces new entities and assumptions in groupoid theory without providing independent evidence or derivations in the abstract; relies on standard background in K-theory and groupoids.

axioms (2)
  • domain assumption Ample groupoids admit discretisation to discrete groupoids under the independent condition.
    Invoked in the definition of independent groupoids and their equivalence.
  • domain assumption Resolutions by independent groupoids preserve homology and K-theory.
    Central to the recipe for building controlled independent resolutions.
invented entities (2)
  • independent groupoids no independent evidence
    purpose: A subclass of ample groupoids equivalent to discrete ones for homology and K-theory.
    New term introduced to enable discretisation; no independent evidence provided in abstract.
  • independent resolutions no independent evidence
    purpose: Sequences of independent groupoids to approximate general ample groupoids.
    New construction for systematic computation; no falsifiable handle outside the paper.

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Works this paper leans on

77 extracted references · 6 canonical work pages

  1. [1]

    Math.,28:1329– 1364, 2022

    C.Antunes, J.Ko, and R.Meyer,The bicategory of groupoid correspondences, New York J. Math.,28:1329– 1364, 2022

    2022

  2. [2]

    G.Arnone, G.Corti ˜nasand D.Mukherjee,Homology of Steinberg algebras, preprint, arXiv:2412.15112

    work page arXiv

  3. [3]

    Math., 167, American Mathematical Society, Providence, RI, 1994

    P.Baum, A.Connesand N.Higson,Classifying space for proper actions and K-theory of groupC ∗-algebras, C∗-algebras: 1943–1993 (San Antonio, TX, 1993), 240–291, Contemp. Math., 167, American Mathematical Society, Providence, RI, 1994

    1943

  4. [4]

    B.Blackadar,K-theory for operator algebras, Second edition, Math. Sci. Res. Inst. Publ., 5, Cambridge University Press, Cambridge, 1998

    1998

  5. [5]

    Math., 372:107314, 73, 2020

    C.B ¨onicke,A going-down principle for ample groupoids and the Baum–Connes conjecture, Adv. Math., 372:107314, 73, 2020

    2020

  6. [6]

    C.B ¨onickeand C.Dell ’ Aiera,Going-down functors and the K¨unneth formula for crossed products by ´etale groupoids, Trans. Amer. Math. Soc.372(2019), no. 11, 8159–8194

    2019

  7. [7]

    C.B ¨onickeand V.Proietti,A categorical approach to the Baum–Connes conjecture for ´etale groupoids, J. Inst. Math. Jussieu23(2024) no. 5, 2319–2364

    2024

  8. [8]

    S.Brown,Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, New York-Berlin, 1982

    K. S.Brown,Cohomology of groups, Graduate Texts in Mathematics, 87, Springer, New York-Berlin, 1982

    1982

  9. [9]

    R.Carri ´on, J.Gabe, C.Schafhauser, A.Tikuisisand S.White,Classifying ∗-homomorphisms I: Unital simple nuclearC ∗-algebras, preprint, arXiv:2307.06480

    J. R.Carri ´on, J.Gabe, C.Schafhauser, A.Tikuisisand S.White,Classifying ∗-homomorphisms I: Unital simple nuclearC ∗-algebras, preprint, arXiv:2307.06480

    work page arXiv

  10. [10]

    Math.6(2001), 127–183

    J.Chabertand S.Echterhoff,Permanence properties of the Baum–Connes conjecture, Doc. Math.6(2001), 127–183

    2001

  11. [11]

    J.Chabert, S.Echterhoffand H.Oyono-Oyono,Going-down functors, the K ¨unneth formula, and the Baum–Connes conjecture, Geom. Funct. Anal.14(2004), no. 3, 491–528

    2004

  12. [12]

    M.Cohenand D

    A. M.Cohenand D. B.Wales,Linearity of Artin groups of finite type, Israel J. Math.131(2002), 101–123

    2002

  13. [13]

    Reine Angew

    M.Crainicand I.Moerdijk,A homology theory for ´etale groupoids, J. Reine Angew. Math.521(2000), 25–46

    2000

  14. [14]

    J.Cuntz, S.Echterhoffand X.Li,On the K-theory of theC ∗-algebra generated by the left regular representation of an Ore semigroup, J. Eur. Math. Soc.17(2015), no. 3, 645–687

    2015

  15. [15]

    J.Cuntz, S.Echterhoffand X.Li,On the K-theory of crossed products by automorphic semigroup actions, Quart. J. Math.64(2013), no. 3, 747–784

    2013

  16. [16]

    J.Cuntz, S.Echterhoff, X.Liand G.Yu,K-Theory for GroupC ∗-Algebras and SemigroupC ∗-Algebras, Oberwolfach Seminars, 47, Birkh¨auser/Springer, Cham, 2017

    2017

  17. [17]

    J.Cuntz, R.Meyerand J.Rosenberg,Topological and bivariant K-theory, Oberwolfach Semin., 36, Birkh¨auser Verlag, Basel, 2007

    2007

  18. [18]

    P.Dehornoy,Foundations of Garside theory, With Franc ¸ois Digne, Eddy Godelle, Daan Krammer and Jean Michel, EMS Tracts in Mathematics, 22, European Mathematical Society, Z¨ urich, 2015

    2015

  19. [19]

    Algebra268(2003), no

    F.Digne,On the linearity of Artin braid groups, J. Algebra268(2003), no. 1, 39–57

    2003

  20. [20]

    Math.12(2019), no

    C.Farsi, A.Kumjian, D.Paskand A.Sims,Ample groupoids: equivalence, homology, and Matui’s HK conjecture, M¨ unster J. Math.12(2019), no. 2, 411–451

    2019

  21. [21]

    2, 159–187

    C.Farthing, P.S.Muhlyand T.Yeend,Higher-rank graphC ∗-algebras: an inverse semigroup and groupoid approach, Semigroup Forum71(2005), no. 2, 159–187

    2005

  22. [22]

    T.Giordano, H.Matui, I.F.Putnamand C.F.Skau,Orbit equivalence for Cantor minimalZ 2-systems, J. Amer. Math. Soc.21(2008), 863–892

    2008

  23. [23]

    Math.179(2010), 119–158

    T.Giordano, H.Matui, I.F.Putnamand C.F.Skau,Orbit equivalence for Cantor minimalZ 𝑑-systems, Invent. Math.179(2010), 119–158

    2010

  24. [24]

    Reine Angew

    T.Giordano, I.F.Putnamand C.F.Skau,Topological orbit equivalence andC ∗-crossed products, J. Reine Angew. Math.469(1995), 51–111

    1995

  25. [25]

    K-Theory9(2024), no

    G.Gong, H.Lin,On classification of nonunital amenable simpleC ∗-algebras IV: Stably projectionlessC ∗- algebras, Ann. K-Theory9(2024), no. 2, 143–339

    2024

  26. [26]

    G.Gong, H.Linand Z.Niu,A classification of finite simple amenableZ-stableC ∗-algebras, I:C ∗-algebras with generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can.42(2020), no. 3, 63–450

    2020

  27. [27]

    G.Gong, H.Linand Z.Niu,A classification of finite simple amenableZ-stableC ∗-algebras, II:C ∗-algebras with rational generalized tracial rank one, C. R. Math. Acad. Sci. Soc. R. Can.42(2020), no. 4, 451–539

    2020

  28. [28]

    E.Guentner, N.Higsonand S.Weinberger,The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes ´Etudes Sci. No.101(2005), 243-–268

    2005

  29. [29]

    Math.225(2021), no

    J.Huangand D.Osajda,Helly meets Garside and Artin, Invent. Math.225(2021), no. 2, 395–426

    2021

  30. [30]

    K.Juschenkoand N.Monod,Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2)178(2013), no. 2, 775–787

    2013

  31. [31]

    Math.6(2000), 1–20

    A.Kumjianand D.Pask,Higher rank graphC ∗-algebras, New York J. Math.6(2000), 1–20

    2000

  32. [32]

    K.Kwa ´sniewskiand R.Meyer,Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness, Doc

    B. K.Kwa ´sniewskiand R.Meyer,Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness, Doc. Math.26(2021), 271–335

    2021

  33. [33]

    X.Li,A new approach to recent constructions ofC ∗-algebras from modular index theory, J. Funct. Anal.269 (2015), 841–864. 52

    2015

  34. [34]

    X.Li,K-theory for semigroupC ∗-algebras and partial crossed products, Comm. Math. Phys.390(2022), no. 1, 1–32

    2022

  35. [35]

    X.Li,Left regular representations of Garside categories I.C ∗-algebras and groupoids, Glasg. Math. J.65 (2023), S53–S86

    2023

  36. [36]

    Finiteness properties of topological full groups, Groups Geom

    X.Li,Left regular representations of Garside categories II. Finiteness properties of topological full groups, Groups Geom. Dyn.20(2026), no. 2, 403–448

    2026

  37. [37]

    Pi.13(2025), Paper No

    X.Li,Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, Forum Math. Pi.13(2025), Paper No. e9, 56 pp

    2025

  38. [38]

    D.Norling,Independent resolutions for totally disconnected dynamical systems I: Algebraic case, J

    X.Liand M. D.Norling,Independent resolutions for totally disconnected dynamical systems I: Algebraic case, J. Algebra424(2015), 98–125

    2015

  39. [39]

    D.Norling,Independent resolutions for totally disconnected dynamical systems

    X.Liand M. D.Norling,Independent resolutions for totally disconnected dynamical systems. II.C ∗-algebraic case, J. Operator Theory75(2016), no.1, 163–193

    2016

  40. [40]

    H.Matui,Homology and topological full groups of ´etale groupoids on totally disconnected spaces, Proc. Lond. Math. Soc.104(2012), 27–56

    2012

  41. [41]

    Reine Angew

    H.Matui,Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math.705(2015), 35–84

    2015

  42. [42]

    Math.303(2016), 502–548

    H.Matui, ´Etale groupoids arising from products of shifts of finite type, Adv. Math.303(2016), 502–548

    2016

  43. [43]

    H.Matui,Classifying Stein’s groups, J. Lond. Math. Soc. (2)112(2025), no. 2, Paper No. e70266, 26 pp

    2025

  44. [44]

    R.Meyer,Homological algebra in bivariant K-theory and other triangulated categories. II.Tbil. Math. J., 1:165–210, 2008

    2008

  45. [45]

    2, 209–259

    R.Meyerand R.Nest,The Baum–Connes conjecture via localisation of categories, Topology45(2006), no. 2, 209–259

    2006

  46. [46]

    I, Triangulated categories, 236–289, London Math

    R.Meyerand R.Nest,Homological algebra in bivariant K-theory and other triangulated categories. I, Triangulated categories, 236–289, London Math. Soc. Lecture Note Ser., 375, Cambridge University Press, Cambridge, 2010

    2010

  47. [47]

    A.Miller,K-theory for ´etale groupoidC ∗-algebras via groupoid correspondences and spectral sequences, PhD Thesis, available athttps://qmro.qmul.ac.uk/xmlui/handle/123456789/83007

    work page arXiv

  48. [48]

    A.Miller,Functors between Kasparov categories from ´etale groupoid correspondences, J. Funct. Anal.287 (2024), no. 11, Paper No. 110623, 48 pp

    2024

  49. [49]

    Noncommut

    A.Miller,Ample groupoid homology and ´etale correspondences, J. Noncommut. Geom.19(2025), no. 2, 657–677

    2025

  50. [50]

    A.Miller,Isomorphisms in K-theory from isomorphisms in groupoid homology theories, Trans. Amer. Math. Soc.378(2025), no. 5, 3349–3391

    2025

  51. [51]

    A.Millerand B.Steinberg,Homology and K-theory for self-similar actions of groups and groupoids, preprint, arXiv:2409.02359

    work page arXiv

  52. [52]

    V.Nekrashevych,Self-similar groups, Math

    V. V.Nekrashevych,Self-similar groups, Math. Surveys Monogr., 117, American Mathematical Society, Providence, RI, 2005

    2005

  53. [53]

    V.Nekrashevych, C ∗-algebras and self-similar groups, J

    V. V.Nekrashevych, C ∗-algebras and self-similar groups, J. Reine Angew. Math.630(2009), 59–123

    2009

  54. [54]

    V.Nekrashevych,Palindromic subshifts and simple periodic groups of intermediate growth, Ann. of Math. (2)187(2018), no. 3, 667–719

    2018

  55. [55]

    D.Norling,The K-theory of some reduced inverse semigroupC ∗-algebras, Math

    M. D.Norling,The K-theory of some reduced inverse semigroupC ∗-algebras, Math. Scand.117(2015), no. 2, 186–202

    2015

  56. [56]

    A. L. T.Paterson,Groupoids, inverse semigroups, and their operator algebras, Progr. Math., 170, Birkh¨auser Boston, Inc., Boston, MA, 1999

    1999

  57. [57]

    Torsion-free ample groupoids, Ergodic Theory Dynam

    V.Proiettiand M.Yamashita,Homology and K-theory of dynamical systems I. Torsion-free ample groupoids, Ergodic Theory Dynam. Systems42(2022), no.8, 2630–2660

    2022

  58. [58]

    V.Proiettiand M.Yamashita,Chern character for torsion-free ample groupoids, preprint, arXiv:2509.07663

    work page arXiv

  59. [59]

    I.Raeburn, A.Simsand T.Yeend,Higher-rank graphs and theirC ∗-algebras, Proc. Edinb. Math. Soc. (2)46 (2003), no. 1, 99–115

    2003

  60. [60]

    I.Raeburn, A.Simsand T.Yeend,TheC ∗-algebras of finitely aligned higher-rank graphs, J. Funct. Anal.213 (2004), no. 1, 206–240

    2004

  61. [61]

    J.55(1987), no

    J.Rosenberg, and C.Schochet,The K ¨unneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J.55(1987), no. 2, 431–474

    1987

  62. [62]

    Topol.11(2007), 1767–1775

    T.Schick,Finite group extensions and the Baum–Connes conjecture, Geom. Topol.11(2007), 1767–1775

    2007

  63. [63]

    R.Skipper, S.Witzeland M.C.B.Zaremsky,Simple groups separated by finiteness properties, Invent. Math. 215(2019), no. 2, 713–740

    2019

  64. [64]

    J.Spielberg,Groupoids andC ∗-algebras for left cancellative small categories, Indiana Univ. Math. J.69 (2020), no. 5, 1579–1626

    2020

  65. [65]

    M.Stein,Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc.332(1992), no. 2, 477–514

    1992

  66. [66]

    Math.223(2010), no

    B.Steinberg,A groupoid approach to discrete inverse semigroup algebras, Adv. Math.223(2010), no. 2, 689–727

    2010

  67. [67]

    Math.216(2019), no

    M.Szymikand N.Wahl,The homology of the Higman–Thompson groups, Invent. Math.216(2019), no. 2, 445–518

    2019

  68. [68]

    O.Tanner,Topological full groups, PhD Thesis, available athttps://theses.gla.ac.uk/84586/. 53

  69. [69]

    O.Tanner,Studying Stein’s Groups as Topological Full Groups, preprint, arXiv:2312.07375

    work page arXiv

  70. [70]

    2, 129–184

    J.-L.Tu,La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory16(1999), no. 2, 129–184

    1999

  71. [71]

    3, 215–264

    J.-L.Tu,La conjecture de Baum–Connes pour les feuilletages moyennables, K-Theory17(1999), no. 3, 215–264

    1999

  72. [72]

    J.-L.Tu,The Baum–Connes Conjecture for groupoids, C ∗-algebras (M¨ unster, 1999), 227-–242, Springer, Berlin, 2000

    1999

  73. [73]

    ETH Z¨ urich, Birkh¨auser Verlag, Basel, 2002

    A.Valette,Introduction to the Baum–Connes conjecture, From notes taken by Indira Chatterji, With an appendix by Guido Mislin, Lectures Math. ETH Z¨ urich, Birkh¨auser Verlag, Basel, 2002

    2002

  74. [74]

    A.Weibel,An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994

    C. A.Weibel,An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994

    1994

  75. [75]

    4, Sections 5–8, 3314–3338, EMS Press, Berlin, 2023

    S.White,Abstract classification theorems for amenableC ∗-algebras, ICM-International Congress of Mathematicians, Vol. 4, Sections 5–8, 3314–3338, EMS Press, Berlin, 2023

    2023

  76. [76]

    P.Williams,Crossed products ofC ∗-algebras, volume 134 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007

    D. P.Williams,Crossed products ofC ∗-algebras, volume 134 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2007

    2007

  77. [77]

    W.Winter,Structure of nuclearC ∗-algebras: from quasidiagonality to classification and back again, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, 1801–1823. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. (X. Li)School of Mathematics and Statistics, University of Glasgow, Unive...

    2018