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The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions

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Pith's one-line read This paper proves that, for a group extension, the Baum–Connes-type assembly maps for the whole group can be reduced to those for the quotient and for finite-subgroup preimages, and gives the first general closure results for the strong Nov

desk verdict Real new SNC/SAC permanence results, but the semidirect-product theorem depends on a sketched K-theory identification and two unsupported claims; referee it and demand details. read the letter →

arxiv 2601.09615 v2 pith:XLV5YVC3 submitted 2026-01-14 math.OA math.KT

Jianguo Zhang This is my paper · ORCID
classification math.OAmath.KT MSC 19K3519K5646L8046L85
keywords Baum–ConnesconjecturestrongNovikovsurjectiveassemblyrationalanalyticgroupextensionsequivariantlocalizationalgebrassemidirectproductsK-theory
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 permanence results for the Baum–Connes and Mishchenko–Kasparov assembly maps with coefficients under group extensions. Its first main theorem says that if every preimage of a finite subgroup of the quotient satisfies the Baum–Connes conjecture with coefficients, and the quotient satisfies the strong Novikov, surjective assembly, and Baum–Connes conjectures with a naturally twisted coefficient algebra, then the whole group satisfies all three conjectures. A second theorem shows that the rational analytic Novikov conjecture can be inherited under a weaker hypothesis: the kernel only needs to satisfy the rational Baum–Connes conjecture. A third theorem treats isometric semidirect products, showing that full conjectures follow from the conjectures for the acting group together with new partial conjectures along the normal subgroup. These closure results yield new examples of groups satisfying the Novikov-type conjectures beyond the class of coarsely embeddable groups.

What carries the argument

The key object is the equivariant localization algebra along P_k(N): a subalgebra of the equivariant localization algebra of P_k(N)×G whose propagation decays only in the N-direction. This algebra interpolates between the full localization algebra of the product and the localization algebra of the quotient, and its K-theory is compared to ordinary localization algebras through a Mayer–Vietoris six-term exact sequence and quantitative K-theory. The two-parametric equivariant localization algebra plays the same interpolating role for isometric semidirect products, while the imprimitivity theorem converts a general group extension into a direct-product situation where these tools apply.

What would settle it

Compute K_*(C*_{L,P_k(N),u}(P_k(N)×G,A)^{N⋊G}) and K_*(C*_{L,P_k(N)}(P_k(N)×G,A)^{N⋊G}) for a concrete isometric semidirect product with nontrivial coefficients and find that they differ; that would invalidate Proposition 6.15 and with it Theorem 6.20. More broadly, a group extension satisfying the hypotheses of Theorem 3.15 whose full assembly map fails injectivity or surjectivity would falsify the permanence claim.

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Extended reading notes

Core claim

The central discovery is that the Baum–Connes assembly map with coefficients is compositional along group extensions: the assembly map for Γ is governed by the assembly maps for the quotient Γ/N and for the preimages q⁻¹(F) of finite subgroups F. Concretely, if each q⁻¹(F) satisfies the Baum–Connes conjecture with coefficients in A, and Γ/N satisfies SNC, SAC, and BCC with coefficients in C₀(Γ/N, A)⋊ᵣΓ, then Γ satisfies the corresponding conjecture with coefficients in A. The paper further shows that for the rational analytic Novikov conjecture, the finite-preimage condition can be weakened to rational Baum–Connes for N, and that for isometric semidirect products the full conjecture follows

Load-bearing premise

The argument rests on the K-theory identification between the uniformly controlled and the ordinary equivariant localization algebras along the normal subgroup; if that identification fails, the assembly map for the whole group cannot be decomposed into the partial maps.

Editorial extensions

If this is right

  • If every finite-subgroup preimage q⁻¹(F) satisfies the Baum–Connes conjecture with coefficients in A and the quotient satisfies SNC, SAC, and BCC with the twisted coefficient algebra C₀(Γ/N, A)⋊ᵣΓ, then Γ satisfies all three conjectures with coefficients in A.
  • If N satisfies the rational Baum–Connes conjecture and Γ/N satisfies the rational analytic Novikov conjecture with twisted coefficients, then Γ satisfies the rational analytic Novikov conjecture.
  • For isometric semidirect products N⋊G, the full Baum–Connes-type conjectures follow from the corresponding conjectures for G with coefficients in certain localization algebras plus the new partial conjectures along N.
  • The strong Novikov conjecture, the surjective assembly conjecture, and the Baum–Connes conjecture with coefficients are closed under direct products, central extensions, and extensions by finite groups.
  • The results produce new groups satisfying the strong and rational analytic Novikov conjectures that are not coarsely embeddable into Hilbert space.

Reading between the lines

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

  • A natural testable extension is to replace the isometric semidirect-product condition by a weaker metric condition on the action of G on N; if the two-parametric localization algebra machinery still yields the K-theory comparison, the same permanence should hold for a broader class of extensions.
  • The rational theorem gives a practical shortcut: for an extension of a finite group, verifying the rational analytic Novikov conjecture only requires rational Baum–Connes for the kernel, even in cases where Baum–Connes with coefficients is known to fail.
  • The quotient-plus-finite-preimages decomposition could be used computationally: higher indices of operators on manifolds with fundamental group Γ might be assembled from computations on the quotient and on finite-subgroup preimages.
  • Because the closure results bypass coarse embeddability, they suggest that the strong Novikov conjecture is more robust under group-theoretic constructions than embeddability itself; similar closure statements might hold for other assembly-type invariants whenever an analogue of the localization-algebra comparison exists.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the Baum–Connes assembly map with coefficients and the Mishchenko–Kasparov assembly map with coefficients for group extensions 1→N→Γ→Γ/N→1. Three main theorems are proved. Theorem 3.15 shows that if q^{-1}(F) satisfies BCC with coefficients in A for every finite subgroup F of Γ/N, and Γ/N satisfies SNC, SAC, and BCC with coefficients in C0(Γ/N,A)⋊rΓ, then Γ satisfies the corresponding conjecture with coefficients in A. Theorem 4.11 gives an analogous statement for the rational analytic Novikov conjecture under the weaker assumption that N satisfies the rational Baum–Connes conjecture. Theorem 6.20 treats isometric semidirect products N⋊G, introducing new partial conjectures along N and reducing SNC/SAC/BCC for N⋊G to the same conjectures for G with coefficients C*_L(P_{k_i}(N),A)^N and the partial conjectures for N⋊G. The paper also draws applications to central extensions, finite extensions, direct products, and constructs examples beyond coarsely embeddable groups.

Significance. The results, if fully established, would give substantial new permanence results for the strong Novikov, surjective assembly, Baum–Connes, and rational analytic Novikov conjectures with coefficients. The paper introduces useful new tools: equivariant localization algebras along one direction, two-parametric equivariant localization algebras, and partial conjectures along N. The proofs use coherent reduction steps via imprimitivity theorems, Rips complex models, and Mayer–Vietoris arguments. The paper also explicitly builds on the author's earlier quantitative K-theory work [46] and product coarse equivalence results [47], which are external and not circular. Examples involving Arzhantseva–Tessera groups are potentially interesting. However, several load-bearing and auxiliary claims are not fully proved, so the current version is not yet at the standard of a definitive journal publication.

major comments (3)
  1. [§6.1, Proposition 6.15] Proposition 6.15 is the load-bearing bridge for Theorem 6.20: it asserts that the embedding τ from the uniformly controlled localization algebra C*_{L,P_k(N),u}(P_k(N)×G,A)^{N⋊G} into the ordinary partial localization algebra C*_{L,P_k(N)}(P_k(N)×G,A)^{N⋊G} induces a K-theory isomorphism. The proof is not written out. The m=0 case is stated, but the induction step is dismissed with 'by a similar argument of the proof of Proposition 5.11'. This comparison is not automatic: the induction in Proposition 5.11 is over skeletons of P_l(G) with stabilizers in G, whereas here the induction is over skeletons of P_k(N) with stabilizers that are finite subgroups of N⋊G, and it must preserve the additional uniform control condition sup_t prop_G(u(t))<∞. The quantitative K-theory facts imported from [46] apply to uniform products of filtered Roe algebras, but the present setting requires demonstratin
  2. [§6.2, Proposition 6.23] Proposition 6.23 is asserted without proof. The text says that 'Kasparov and Yu's method in [24] is also practicable for the (rational) strong Novikov conjecture along N with coefficients' and then states the proposition. This proposition is used to obtain Corollary 6.24 and Examples 7.17 and 7.18. As a new result, it requires a proof; otherwise a precise reference to a source where the coefficient-version of the Kasparov–Yu argument is established is necessary. As written, this is an unsupported assertion.
  3. [§7.4, Example 7.19] Example 7.19 relies on an unstated generalization of [13, Proposition 8.8] to arbitrary coefficients. The text asserts that this generalization holds 'by a Mayer–Vietoris argument and the Künneth formula, just like the proof of Lemma 4.1', but no details are given. Since the example is used to claim that certain extensions satisfy the rational analytic Novikov conjecture with coefficients, this missing support should be supplied or the example should be downgraded to a conditional statement.
minor comments (5)
  1. [§3.2, Theorem 3.15] In condition (2), the coefficient algebra C0(Γ/N,A)⋊rΓ must be viewed as a Γ/N-C*-algebra for the conjectures for Γ/N to be stated. The Γ/N-action is only implicit from the action β' in (3.2). Please spell out the action explicitly to avoid ambiguity.
  2. [§5.3, Proposition 5.11] In the n=0 case of the proof, the displayed identification of the left-hand side of ψ_* is written with a tensor product arrangement that is not immediately clear. Please expand the identification of the coefficient algebra and the rôle of the unitary V.
  3. [§7.3, Example 7.17] The example says the direct product satisfies the strong Novikov conjecture by Corollary 7.10. As stated, Corollary 7.10 requires BCC with coefficients for both factors, which is not known for all coarsely embeddable groups. The intended conclusion can be obtained by applying only the SNC component of Theorem 7.9, but the citation to Corollary 7.10 is misleading and should be clarified.
  4. [§4.1, Lemma 4.1] The proof invokes an invariant trace τ_ΩG on C(Ω_G) and uses it to define a trace on B=C(Ω_G)⋊rG. A reference or construction for this trace would be helpful, as it is not immediate.
  5. [§2.4, Definition 2.21] In the definition of the Milnor–Rips complex, the equivalence relation in condition (3b) could be phrased more clearly, and the index in the formula for λ in (2.1) should be checked against the definition of the equivalence classes.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorems reduce assembly maps to independent factor conjectures; author self-citations are technical tools, not encoded conclusions.

full rationale

The derivation chain is not circular. The main assembly maps are defined as limits of K-theory of localization and Roe algebras; no fitted parameters appear and no target conjecture is inserted as an assumption in disguised form. Theorem 3.15 uses Green imprimitivity and a direct-product reduction: hypothesis (1) concerns finite preimages q^{-1}(F), hypothesis (2) concerns the quotient Γ/N, and neither is equivalent to the conclusion for Γ. Theorem 4.11 similarly reduces RANC for Γ to rational BCC for N and RANC for Γ/N via one-way imprimitivity lemmas; Lemma 4.7 is not inverted to obtain the conclusion from itself. Theorem 6.20 is the only place where a new 'partial conjecture along N' appears, but it is not a renaming of the full conjecture: Proposition 5.12 factors the full assembly map for N⋊G as (5.6) followed by (5.8), and Proposition 6.18 identifies (5.8) with the partial assembly map (6.2). Thus the theorem is a genuine composition-type reduction, not an identity. The self-citations [46] and [47] supply quantitative K-theory and Rips-complex homotopy equivalences; these are external technical lemmas with independent statements, not definitions of the conjectures, so they do not make the argument circular. The most notable gap is the proof of Proposition 6.15, where the induction step is dismissed with 'by a similar argument of the proof of Proposition 5.11'; this is a rigor concern, not a circularity, and does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 4 invented entities

The paper introduces no fitted constants. Its central claims rest on standard tools (imprimitivity, Mayer-Vietoris, quantitative K-theory), on two self-cited technical papers [46, 47], and on one unproved extension of the Kasparov-Yu method (Proposition 6.23). The main theorems are not forced by construction, but the ledger contains several substantial imported or asserted premises.

assumptions (6)
  • domain assumption Künneth formula in K-theory for the coefficient algebra A is assumed in Theorem 4.11 and Lemma 4.1.
    Stated in Section 4 before Lemma 4.1 and in Theorem 4.11; used to construct rational Künneth isomorphisms for localization algebra K-theory.
  • standard math Green's imprimitivity theorem and the induction equivalence between K-theory of localization algebras for a subgroup and the induced algebra.
    Cited as [7, Theorem 2.2] and [14, Theorem 17]; used in Proposition 3.11 and Lemma 4.7.
  • ad hoc to paper The Kasparov-Yu Property (H) coarse embedding argument applies to the newly defined partial assembly maps along N.
    Proposition 6.23 is asserted without proof; it is used in Corollary 6.24 and the examples of Section 7.3.
  • standard math Quantitative K-theory lemmas (Lemma 6.9, 6.13, 6.14) from the author's prior paper [46] hold for the constructed filtered algebras.
    Cited from [46]; used in the proof of Proposition 6.15 to compare uniformly controlled and ordinary localization algebras.
  • standard math P_k(N×G) is strongly Lipschitz and coarsely equivalent to P_k(N)×P_k(G).
    Taken from [47, Lemma 4.17]; used in Lemma 3.1 and Lemma 5.3 to replace the Rips complex of a product or semidirect product by a product of Rips complexes.
  • standard math The ordering maps and trace on C(Ω_G)⋊rG from [17] have the stated properties.
    Used in the proof of Lemma 4.1 to establish rational injectivity of λ* for product Milnor-Rips complexes.
invented entities (4)
  • Equivariant localization algebra along P_k(N) with coefficients (Definition 3.3)
    purpose: Decompose the Baum-Connes map for direct products into a G-part and an N-part.
    New C*-algebra defined for the proof; its K-theory is shown isomorphic to known localization algebras only under the hypotheses of Propositions 3.6 and 3.8.
  • Two-parametric equivariant localization algebra C*_LL(P_k(N)×P_l(G),A)^{N⋊G} (Definition 5.5)
    purpose: Bridge the localization algebra of N⋊G with the iterated algebra C*_L(P_l(G), C*_L(P_k(N),A)^N)^G.
    Instrumental in Theorem 6.20; no external falsifiable prediction is attached to it.
  • Uniformly controlled equivariant localization algebra along P_k(N) (Definition 6.1)
    purpose: Show that the partial assembly maps along N have the same K-theory as the full localization algebra along P_k(N).
    Defined in Section 6; the K-theory isomorphism with the ordinary localization algebra is the content of Proposition 6.15.
  • Conjectures 'along N' (SNC, SAC, BCC, RANC along N, Conjecture 6.19)
    purpose: Formulate hypotheses for semidirect products that are weaker than full Baum-Connes for fibers.
    New conjectural statements introduced by the paper; used in Theorems 6.20 and 6.22.

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Pith. "Pith review of The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions." pith.science (2026). https://pith.science/paper/XLV5YVC3

@misc{pith2026260109615,
  author       = {Pith},
  title        = {Pith review of: The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLV5YVC3}},
  note         = {Machine review of arXiv:2601.09615}
}
abstract

In this paper, we investigate the injectivity, surjectivity and isomorphism of the Baum--Connes assembly map $e_{\ast}$ with coefficients, and the injectivity of the Mishchenko--Kasparov assembly map $\mu_{\ast}$ with coefficients for group extensions $1\rightarrow N \rightarrow \Gamma \xrightarrow{q} \Gamma/ N \rightarrow 1$. The main results are as follows. (1) Under the assumption that $e_{\ast}$ is isomorphic for $q^{-1}(F)$ for any finite subgroup $F$ of $\Gamma/N$, we prove that $e_{\ast}$ is injective, surjective and isomorphic for $\Gamma$ if they are also true for $\Gamma/N$, respectively. (2) Under the assumption that $e_{\ast}$ is rationally isomorphic for $N$, we verify that $\mu_{\ast}$ is rationally injective for $\Gamma$ if it is also rationally injective for $\Gamma/N$. (3) When $\Gamma$ is an isometric semi-direct product $N\rtimes G$, we confirm that $e_{\ast}$ is injective, surjective and isomorphic for $\Gamma$ if they also hold for $G$ and $\Gamma$ satisfies three partial conjectures along $N$, respectively. As applications, we show that the strong Novikov conjecture, the surjective assembly conjecture and the Baum--Connes conjecture with coefficients are closed under direct products, central extensions of groups and extensions by finite groups. Meanwhile, we also show that the rational analytic Novikov conjecture with coefficients is preserved under extensions of finite groups. Besides, we employ these results to obtain some new examples for the rational analytic and the strong Novikov conjecture beyond the class of coarsely embeddable groups.

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Reference graph

Works this paper leans on

47 extracted references · 3 linked inside Pith

  1. [46]

    On the quantitative coarse Baum–Connes conjecture with coefficients

    Jianguo Zhang. On the quantitative coarse Baum–Connes conjecture with coefficients. 2024. arXiv:2410.11929, to appear in Sci. China Math

    arXiv 2024

  2. [47]

    The coarse Baum–Connes conjecture with filtered coefficients and product metric spaces.Adv

    Jianguo Zhang. The coarse Baum–Connes conjecture with filtered coefficients and product metric spaces.Adv. Math., 475:Paper No. 110327, 2025. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China. Email address:jgzhang@snnu.edu.cn 42

    2025

  3. [24]

    The Novikov conjecture and geometry of Banach spaces.Geom

    Gennadi Kasparov and Guoliang Yu. The Novikov conjecture and geometry of Banach spaces.Geom. Topol., 16(3):1859–1880, 2012

    2012

  4. [1]

    BivariantK-theory with R/Z-coefficients and rho classes of unitary representations.J

    Paolo Antonini, Sara Azzali, and Georges Skandalis. BivariantK-theory with R/Z-coefficients and rho classes of unitary representations.J. Funct. Anal., 270(1):447–481, 2016

    2016

  5. [2]

    The Baum–Connes con- jecture localised at the unit element of a discrete group.Compos

    Paolo Antonini, Sara Azzali, and Georges Skandalis. The Baum–Connes con- jecture localised at the unit element of a discrete group.Compos. Math., 156(12):2536–2559, 2020

    2020

  6. [3]

    A categorical perspective on the Atiyah– Segal completion theorem in KK-theory.J

    Yuki Arano and Yosuke Kubota. A categorical perspective on the Atiyah– Segal completion theorem in KK-theory.J. Noncommut. Geom., 12(2):779– 821, 2018

    2018

  7. [4]

    Admitting a coarse embedding is not preserved under group extensions.Int

    Goulnara Arzhantseva and Romain Tessera. Admitting a coarse embedding is not preserved under group extensions.Int. Math. Res. Not. IMRN, (20):6480– 6498, 2019

    2019

  8. [5]

    Classifying space for proper actions andK-theory of groupC ∗-algebras

    Paul Baum, Alain Connes, and Nigel Higson. Classifying space for proper actions andK-theory of groupC ∗-algebras. InC ∗-algebras: 1943–1993 (San Antonio, TX, 1993), volume 167 ofContemp. Math., pages 240–291. Amer. Math. Soc., Providence, RI, 1994

    1943

Show all 47 references
  1. [6]

    Boundary amenabil- ity of Out(F N ).Ann

    Mladen Bestvina, Vincent Guirardel, and Camille Horbez. Boundary amenabil- ity of Out(F N ).Ann. Sci. ´Ec. Norm. Sup´ er. (4), 55(5):1379–1431, 2022

    2022

  2. [7]

    Permanence properties of the Baum– Connes conjecture.Doc

    J´ erˆ ome Chabert and Siegfried Echterhoff. Permanence properties of the Baum– Connes conjecture.Doc. Math., 6:127–183, 2001

    2001

  3. [8]

    Going-down functors, the K¨ unneth formula, and the Baum–Connes conjecture.Geom

    J´ erˆ ome Chabert, Siegfried Echterhoff, and Herv´ e Oyono-Oyono. Going-down functors, the K¨ unneth formula, and the Baum–Connes conjecture.Geom. Funct. Anal., 14(3):491–528, 2004

    2004

  4. [9]

    Constructions preserving Hilbert space uniform embeddability of discrete groups.Trans

    Marius Dadarlat and Erik Guentner. Constructions preserving Hilbert space uniform embeddability of discrete groups.Trans. Amer. Math. Soc., 355(8):3253–3275, 2003

    2003

  5. [10]

    The Novikov conjecture and extensions of coarsely embeddable groups.J

    Jintao Deng. The Novikov conjecture and extensions of coarsely embeddable groups.J. Noncommut. Geom., 16(1):265–310, 2022

    2022

  6. [11]

    The Baum– Connes conjecture: an extended survey

    Maria Paula Gomez Aparicio, Pierre Julg, and Alain Valette. The Baum– Connes conjecture: an extended survey. InAdvances in noncommutative geometry—on the occasion of Alain Connes’ 70th birthday, pages 127–244. Springer, Cham, [2019]©2019

    2019

  7. [12]

    The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert– Hadamard spaces

    Sherry Gong, Jianchao Wu, Zhizhang Xie, and Guoliang Yu. The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert– Hadamard spaces. 2023. arXiv:2310.01219

    2023 arXiv

  8. [13]

    The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert–Hadamard spaces

    Sherry Gong, Jianchao Wu, and Guoliang Yu. The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert–Hadamard spaces. Geom. Funct. Anal., 31(2):206–267, 2021

    2021

  9. [14]

    The local structure of twisted covariance algebras.Acta Math., 140(3-4):191–250, 1978

    Philip Green. The local structure of twisted covariance algebras.Acta Math., 140(3-4):191–250, 1978. 40

    1978

  10. [15]

    The Novikov conjecture for linear groups.Publ

    Erik Guentner, Nigel Higson, and Shmuel Weinberger. The Novikov conjecture for linear groups.Publ. Math. Inst. Hautes ´Etudes Sci., (101):243–268, 2005

    2005

  11. [16]

    Dynamical complexity and controlled operatorK-theory.Ast´ erisque, (451):89, 2024

    Erik Guentner, Rufus Willett, and Guoliang Yu. Dynamical complexity and controlled operatorK-theory.Ast´ erisque, (451):89, 2024

    2024

  12. [17]

    Hilbert–Hadamard spaces and the equivariant coarse Novikov conjecture

    Liang Guo, Qin Wang, Jianchao Wu, and Guoliang Yu. Hilbert–Hadamard spaces and the equivariant coarse Novikov conjecture. 2024. arXiv:2411.18538

    2024 arXiv

  13. [18]

    Geometry of the mapping class groups

    Ursula Hamenst¨ adt. Geometry of the mapping class groups. I. Boundary amenability.Invent. Math., 175(3):545–609, 2009

    2009

  14. [19]

    Math., 144(1):23–74, 2001

    Nigel Higson and Gennadi Kasparov.E-theory andKK-theory for groups which act properly and isometrically on Hilbert space.Invent. Math., 144(1):23–74, 2001

    2001

  15. [20]

    Counterexamples to the Baum–Connes conjecture.Geom

    Nigel Higson, Vincent Lafforgue, and Georges Skandalis. Counterexamples to the Baum–Connes conjecture.Geom. Funct. Anal., 12(2):330–354, 2002

    2002

  16. [21]

    EquivariantKK-theory and the Novikov conjecture.In- vent

    Gennadi Kasparov. EquivariantKK-theory and the Novikov conjecture.In- vent. Math., 91(1):147–201, 1988

    1988

  17. [22]

    InNovikov conjectures, index theorems and rigidity, Vol

    Gennadi Kasparov.K-theory, groupC ∗-algebras, and higher signatures (con- spectus). InNovikov conjectures, index theorems and rigidity, Vol. 1 (Ober- wolfach, 1993), volume 226 ofLondon Math. Soc. Lecture Note Ser., pages 101–146. Cambridge Univ. Press, Cambridge, 1995

    1993

  18. [23]

    Groups acting properly on “bolic” spaces and the Novikov conjecture.Ann

    Gennadi Kasparov and Georges Skandalis. Groups acting properly on “bolic” spaces and the Novikov conjecture.Ann. of Math. (2), 158(1):165–206, 2003

    2003

  19. [25]

    The mapping class group from the viewpoint of measure equivalence theory.Mem

    Yoshikata Kida. The mapping class group from the viewpoint of measure equivalence theory.Mem. Amer. Math. Soc., 196(916):viii+190, 2008

    2008

  20. [26]

    Math., 149(1):1–95, 2002

    Vincent Lafforgue.K-th´ eorie bivariante pour les alg` ebres de Banach et con- jecture de Baum–Connes.Invent. Math., 149(1):1–95, 2002

    2002

  21. [27]

    La conjecture de Baum–Connes ` a coefficients pour les groupes hyperboliques.J

    Vincent Lafforgue. La conjecture de Baum–Connes ` a coefficients pour les groupes hyperboliques.J. Noncommut. Geom., 6(1):1–197, 2012

    2012

  22. [28]

    The Baum–Connes conjecture for extensions.M¨ unster J

    Ralf Meyer. The Baum–Connes conjecture for extensions.M¨ unster J. Math., 18(1):245–247, 2025

    2025

  23. [29]

    The Baum–Connes conjecture via localisation of categories.Topology, 45(2):209–259, 2006

    Ralf Meyer and Ryszard Nest. The Baum–Connes conjecture via localisation of categories.Topology, 45(2):209–259, 2006

    2006

  24. [30]

    The Baum–Connes conjecture for hyperbolic groups.Invent

    Igor Mineyev and Guoliang Yu. The Baum–Connes conjecture for hyperbolic groups.Invent. Math., 149(1):97–122, 2002

    2002

  25. [31]

    Nowak and Guoliang Yu.Large scale geometry

    Piotr W. Nowak and Guoliang Yu.Large scale geometry. EMS Textbooks in Mathematics. EMS Press, Berlin, second edition, [2023]©2023

    2023

  26. [32]

    Baum–Connes conjecture and extensions.J

    Herv´ e Oyono-Oyono. Baum–Connes conjecture and extensions.J. Reine Angew. Math., 532:133–149, 2001

    2001

  27. [33]

    On quantitative operatorK-theory

    Herv´ e Oyono-Oyono and Guoliang Yu. On quantitative operatorK-theory. Ann. Inst. Fourier (Grenoble), 65(2):605–674, 2015

    2015

  28. [34]

    QuantitativeK-theory and the K¨ unneth formula for operator algebras.J

    Herv´ e Oyono-Oyono and Guoliang Yu. QuantitativeK-theory and the K¨ unneth formula for operator algebras.J. Funct. Anal., 277(7):2003–2091, 2019

    2003

  29. [35]

    On the localization algebra of Guoliang Yu.Forum Math., 22(4):657–665, 2010

    Yu Qiao and John Roe. On the localization algebra of Guoliang Yu.Forum Math., 22(4):657–665, 2010

    2010

  30. [36]

    An index theorem on open manifolds

    John Roe. An index theorem on open manifolds. I, II.J. Differential Geom., 27(1):87–113, 115–136, 1988

    1988

  31. [37]

    Hautes ´Etudes Sci

    Jonathan Rosenberg.C ∗-algebras, positive scalar curvature, and the Novikov conjecture.Inst. Hautes ´Etudes Sci. Publ. Math., (58):197–212, 1983

    1983

  32. [38]

    Topological methods forC ∗-algebras

    Claude Schochet. Topological methods forC ∗-algebras. II. Geometric resolu- tions and the K¨ unneth formula.Pacific J. Math., 98(2):443–458, 1982. 41

    1982

  33. [39]

    Uniform embeddings of hyperbolic groups in Hilbert spaces.Israel J

    Zlil Sela. Uniform embeddings of hyperbolic groups in Hilbert spaces.Israel J. Math., 80(1-2):171–181, 1992

    1992

  34. [40]

    Georges Skandalis, Jean-Louis Tu, and Guoliang. Yu. The coarse Baum– Connes conjecture and groupoids.Topology, 41(4):807–834, 2002

    2002

  35. [41]

    Cambridge University Press, Cambridge, 2020

    Rufus Willett and Guoliang Yu.Higher index theory, volume 189 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020

    2020

  36. [42]

    Baum–Connes conjecture and coarse geometry.K-Theory, 9(3):223–231, 1995

    Guoliang Yu. Baum–Connes conjecture and coarse geometry.K-Theory, 9(3):223–231, 1995

    1995

  37. [43]

    Localization algebras and the coarse Baum–Connes conjecture

    Guoliang Yu. Localization algebras and the coarse Baum–Connes conjecture. K-Theory, 11(4):307–318, 1997

    1997

  38. [44]

    The Novikov conjecture for groups with finite asymptotic di- mension.Ann

    Guoliang Yu. The Novikov conjecture for groups with finite asymptotic di- mension.Ann. of Math. (2), 147(2):325–355, 1998

    1998

  39. [45]

    The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space.Invent

    Guoliang Yu. The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space.Invent. Math., 139(1):201–240, 2000

    2000

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