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Advances in Noncommutative Geometry pp 127–244Cite as

The Baum–Connes conjecture: an extended survey

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Abstract

We present a history of the Baum–Connes conjecture, the methods involved, the current status, and the mathematics it generated.

To Alain Connes, for providing lifelong inspiration

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Notes

  1. 1.

    This is important, e.g., for applications to foliations, see Chapter 7.

  2. 2.

    In the case of SL 3(Z), surjectivity of μ r is the open problem; the LHS of the Baum–Connes conjecture was computed in [SG08].

  3. 3.

    Recall that B(d, n) is defined as the quotient of the non-abelian free group F d by the normal subgroup generated by all n’s powers in F d.

  4. 4.

    Compare with Sections 2.5 and 4.5.

  5. 5.

    G∕K carries a G-invariant spin structure if and only if ρ − ρ c ∈ Λ, see [AS77, 4.34]; the distinction between S + and S − is made by requiring that ρ − ρ c is the highest weight for S +, see [AS77, 3.13].

  6. 6.

    We believe that Connes and Moscovici actually had \(C^{\ast }_r(G)\), not C ∗(G), in mind when writing this.

  7. 7.

    As Connes once pointed out: “E Γ is a point on which Γ acts freely!.”

  8. 8.

    Although published only in 1995, the celebrated “ Conspectus” was first circulated in 1981.

  9. 9.

    K-homology is the homology theory dual to topological K-theory. It was shown by Atiyah [Ati70] that an elliptic (pseudo-)differential operator on M defines an element in K 0(M).

  10. 10.

    When B Γ is a general CW-complex we must replace K 0(B Γ) by , where X runs along compact subsets of B Γ.

  11. 11.

    In terms of Kasparov theory, to be defined in Chapter 3 below, this can be expressed using Kasparov product: \(\beta [D]=[\mathcal {L}_\Gamma ]\otimes _{C(B\Gamma )}[D]\).

  12. 12.

    The injectivity of \(\tilde \alpha _A\) is responsible for the Novikov conjecture, Conjecture 3: see Section 4.5.1.

  13. 13.

    This is actually the same map as the map β from Section 2.5.

  14. 14.

    The fact that it is indeed an equivalence relation does not appear in [BC00].

  15. 15.

    Note that the proof is given there only for discrete groups, but the proof goes over to locally compact group.

  16. 16.

    See also Higson and Guentner [HG04, Theorem 2.19] and Kasparov and Skandalis [KS03]. The case where G is a connected Lie group and B = C 0(X), where X is a proper G-space, was previously treated by Valette [Val88].

  17. 17.

    For a nice proof of that result NOT appealing to Atiyah’s L 2-index theorem, see lemma 7.1 in [MV03].

  18. 18.

    Or is a-(T)-menable, according to Gromov.

  19. 19.

    A concrete example of a non-cocompact lattice in Sp(n,  1), is Sp(n,  1)(H(Z)), the group of points of the real algebraic group Sp(n,  1) over the ring H(Z) of integral quaternions. For such a group Conjecture 5 is still open.

  20. 20.

    See the discussion of strong property (T) in 6.3.

  21. 21.

    Until the end of Proposition 7.6, we denote a countable group by Γ∞ rather than Γ, as we view Γ∞ as the limit of its finite quotients Γk.

  22. 22.

    If Γ∞ has property (T), e π is the image in \(C^{\ast }_\pi (\Gamma _\infty )\) of the Kazhdan projection \(e_{\mathcal {G}}\in C^{\ast }_{\mathrm {max}}(\Gamma _\infty )\) from Proposition 5.4.

  23. 23.

    The re-interpretation goes as follows: fix an auxiliary orientation on the edges of E, allowing one to define the coboundary operator d : ℓ 2(V ) → ℓ 2(E) : ϕ↦dϕ, where dϕ(e) = ϕ(e +) − ϕ(e −). Observe that Δ = d ∗d, so that \(\langle \Delta \phi ,\phi \rangle =\|d\phi \|{ }^2=\frac {1}{2}\sum _{x\sim y}|\phi (x)-\phi (y)|{ }^2\). By the Rayleigh quotient, \(\frac {1}{\lambda _1}\) is the smallest constant K > 0 such that ∥ϕ∥2 ≤ K∥dϕ∥2 for every ϕ ⊥ 1. We leave the rest as an exercise.

  24. 24.

    Recall that a group acting properly isometrically on a CAT(0) cube complex, has the Haagerup property, see, e.g., Corollary 1 in [Val18].

  25. 25.

    Amenability (resp. property (T)) can be defined by a fixed point property: existence of a fixed point for affine actions on compact convex sets (resp. affine isometric actions on Hilbert spaces). This makes clear that it is preserved under extensions.

  26. 26.

    Under the assumption that | Γ| coarsely embeds into Hilbert space, the assumption that B Γ is a finite complex was removed by Skandalis et al. [STY02], using their groupoid approach to CBC.

  27. 27.

    Recall from Section 8.2.3 that an entourage is a subset of X × X on which d(., .) is bounded.

  28. 28.

    Note typos regarding inequality 9.3 in Proposition 5.2 of [Laf08] and in Proposition 5.5 of [Laf09]: \(\leq \frac {4}{|X_k|}\) is erroneously written as \(=\frac {2}{|X_k|}\).

  29. 29.

    The subtlety here is that, as lucidly explained in [dlS16], Definition 6.32 for an arbitrary finitely generated group is equivalent to the existence of a sequence of signed probability measures as above.

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Acknowledgement

Thanks are due to J.-B. Bost, R. Coulon, N. Higson, V. Lafforgue, P.-Y. Le Gall, H. Oyono-Oyono, N. Ozawa, and M. de la Salle for useful conversations and exchanges.

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  1. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France

    Maria Paula Gomez Aparicio

  2. Institut Denis Poisson, Université d’Orléans, Collegium Sciences et Techniques, Bâtiment de mathématiques, Orléans, France

    Pierre Julg

  3. Institut de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland

    Alain Valette

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    Guoliang Yu

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Aparicio, M.P.G., Julg, P., Valette, A. (2019). The Baum–Connes conjecture: an extended survey. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_3

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