The Bost conjecture predicts that for every countable discrete group GG the assembly map μG ⁣:K∗G(E‾G)→K∗top(ℓ1(G))\mu^G\colon K^G_*(\underline{E}G)\to K^{\mathrm{top}}_*(\ell^1(G)), from the equivariant K-homology of the classifying space for proper GG-actions to the topological K-theory of the Banach algebra ℓ1(G)\ell^1(G), is an isomorphism. Berrick, Chatterji and Mislin proved that the Bost conjecture implies the classical Bass conjecture on Hattori–Stallings ranks of finitely generated projective ZG\mathbb{Z}G-modules, and consequently that all amenable groups satisfy the Bass conjecture. The Bost conjecture itself remains open for general countable discrete groups: it is known for a-T-menable (Haagerup) groups, word-hyperbolic groups, lattices in reductive Lie groups, and related geometric classes, and no counterexample is known.

Background

For a discrete group GG and a commutative ring RR, the group ring RGRG consists of finite formal RR-linear combinations of group elements, with multiplication induced by the group law. A finitely generated projective RGRG-module is a direct summand of a finite free RGRG-module. The Hattori–Stallings rank of such a module is its trace, valued in the quotient of RGRG by the subspace spanned by commutators gh−hggh-hg, which identifies with the free module on the conjugacy classes of GG (over a field of characteristic 00). The classical Bass conjecture asserts that for a finitely generated projective ZG\mathbb{Z}G-module the Hattori–Stallings rank is supported on the conjugacy class of the identity element; equivalently, the trace vanishes on all non-identity conjugacy classes. It is a finiteness statement about the integral group ring that controls, for example, which idempotents can exist in matrix algebras over CG\mathbb{C}G.

On the analytic side, ℓ1(G)\ell^1(G) is the Banach algebra of absolutely summable complex-valued functions on GG with convolution product; it embeds continuously into the reduced group C∗C^*-algebra Cr∗(G)C^*_r(G), the completion of the complex group ring CG\mathbb{C}G for its action by left convolution on the Hilbert space ℓ2(G)\ell^2(G). Topological K-theory K∗top(ℓ1(G))K^{\mathrm{top}}_*(\ell^1(G)) is defined as for Banach algebras generally: K0K_0 is the Grothendieck group of finitely generated projective ℓ1(G)\ell^1(G)-modules, and the higher groups are built from homotopy classes of invertible matrices over ℓ1(G)\ell^1(G) and over its suspensions. The Baum–Connes conjecture with trivial coefficients predicts that an assembly map from the equivariant K-homology K∗G(E‾G)K^G_*(\underline{E}G) — where E‾G\underline{E}G denotes the classifying space for proper GG-actions, a proper GG-CW-complex universal among such spaces — to K∗(Cr∗(G))K_*(C^*_r(G)) is an isomorphism; the Bost conjecture is the variant of this statement with the Banach algebra ℓ1(G)\ell^1(G) in place of the reduced C∗C^*-algebra. The Baum–Connes assembly map factors through the Bost assembly map: it is the composite of μG\mu^G with the map on K-theory induced by the inclusion ℓ1(G)↪Cr∗(G)\ell^1(G)\hookrightarrow C^*_r(G) (Lafforgue; Gómez Aparicio–Julg–Valette). Consequently, the Bost conjecture implies the Baum–Connes conjecture exactly for those groups for which this inclusion induces an isomorphism in K-theory. That comparison holds whenever ℓ1(G)\ell^1(G) is closed under holomorphic functional calculus in Cr∗(G)C^*_r(G), by the classical density theorem for such dense subalgebras (Swan, Karoubi, Bost), but it is itself open in general: ℓ1(G)\ell^1(G) fails to be stable under holomorphic calculus already when GG is a non-compact semisimple Lie group, and the Bost conjecture is known for lattices in reductive Lie groups, such as SL3(Z)\mathrm{SL}_3(\mathbb{Z}), for which the Baum–Connes conjecture remains open.

The ℓ1\ell^1-formulation is more flexible than the C∗C^*-formulation for homological arguments. Berrick, Chatterji and Mislin (Math. Ann. 329, 2004) used this flexibility to prove that the Bost conjecture implies the classical Bass conjecture; combined with Lafforgue's bivariant K-theory for Banach algebras (Invent. Math. 149, 2002), which establishes the conjecture for a-T-menable groups — groups admitting a proper affine isometric action on Hilbert space, also called groups with the Haagerup property, a class that contains all amenable groups (groups admitting a left-invariant finitely additive probability measure on all subsets) — this yields the Bass conjecture for all amenable groups. Lafforgue's methods also cover word-hyperbolic groups (finitely generated groups whose Cayley graph is negatively curved in Gromov's sense) and lattices in reductive Lie groups; the Novikov conjecture for word-hyperbolic groups — the assertion that the higher signatures of closed oriented manifolds are invariant under orientation-preserving homotopy equivalences — had been proved earlier by Connes and Moscovici via cyclic cohomology (Topology 29, 1990), and Higson and Kasparov (Invent. Math. 144, 2001) proved the corresponding C∗C^*-assembly isomorphism for a-T-menable groups. Later work has extended the known classes further — to open subgroups and groups acting on trees, and to proper Banach algebra coefficients (Paravicini, J. K-Theory 4, 2009 and J. Noncommut. Geom. 7, 2013) — but always within geometrically or structurally restricted families. The Bost conjecture for arbitrary countable discrete groups — the hard cases being groups fitting none of the known geometric patterns, for instance many groups with Kazhdan's property (T), a representation-theoretic rigidity property that rules out proper affine isometric actions on Hilbert space, that do not arise as lattices in Lie groups — remains open, and no counterexample is known (Gómez Aparicio–Julg–Valette, 2019).

Problem Statement

Is the Bost assembly map μG ⁣:K∗G(E‾G)⟶K∗top(ℓ1(G))\mu^G\colon K^G_*(\underline{E}G)\longrightarrow K^{\mathrm{top}}_*(\ell^1(G)) an isomorphism for every countable discrete group GG? Here E‾G\underline{E}G is the classifying space for proper GG-actions, K∗G(E‾G)K^G_*(\underline{E}G) its equivariant K-homology, ℓ1(G)\ell^1(G) the convolution Banach algebra of absolutely summable functions on GG, and K∗top(ℓ1(G))K^{\mathrm{top}}_*(\ell^1(G)) the topological K-theory of that Banach algebra. The task is to resolve this question: either prove that μG\mu^G is an isomorphism for every countable discrete group GG, or exhibit a countable discrete group GG for which μG\mu^G fails to be an isomorphism. By the theorem of Berrick–Chatterji–Mislin, an affirmative answer would in particular settle the classical Bass conjecture (vanishing of the Hattori–Stallings rank off the identity conjugacy class) for all finitely generated projective ZG\mathbb{Z}G-modules, for every group GG.

References

  1. A. J. Berrick, I. Chatterji, G. Mislin, "From acyclic groups to the Bass conjecture for amenable groups", Mathematische Annalen 329 (2004), 597–621, DOI 10.1007/s00208-004-0521-6, https://doi.org/10.1007/s00208-004-0521-6 (also arXiv:1004.1941, https://arxiv.org/abs/1004.1941)
  2. A. Connes, H. Moscovici, "Cyclic cohomology, the Novikov conjecture and hyperbolic groups", Topology 29 (1990), 345–388, DOI 10.1016/0040-9383(90)90003-3, https://doi.org/10.1016/0040-9383(90)90003-3
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  5. V. Lafforgue, "K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes", Inventiones mathematicae 149 (2002), 1–95, DOI 10.1007/s002220200213, https://doi.org/10.1007/s002220200213
  6. W. Paravicini, "The Bost conjecture, open subgroups and groups acting on trees", Journal of K-Theory 4 (2009), 469–490, DOI 10.1017/is009010006jkt074, https://doi.org/10.1017/is009010006jkt074 (also arXiv:0902.4339, https://arxiv.org/abs/0902.4339)
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