The Bost conjecture: is the $\ell^1$-assembly map an isomorphism for every countable discrete group?

The Bost conjecture predicts that for every countable discrete group $G$ the assembly map $\mu^G\colon K^G_*(\underline{E}G)\to K^{\mathrm{top}}_*(\ell^1(G))$, from the equivariant K-homology of the classifying space for proper $G$-actions to the topological K-theory of the Banach algebra $\ell^1(G)$, 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 $\mathbb{Z}G$-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 $G$ and a commutative ring $R$, the group ring $RG$ consists of finite formal $R$-linear combinations of group elements, with multiplication induced by the group law. A finitely generated projective $RG$-module is a direct summand of a finite free $RG$-module. The Hattori–Stallings rank of such a module is its trace, valued in the quotient of $RG$ by the subspace spanned by commutators $gh-hg$, which identifies with the free module on the conjugacy classes of $G$ (over a field of characteristic $0$). The classical Bass conjecture asserts that for a finitely generated projective $\mathbb{Z}G$-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 $\mathbb{C}G$.

On the analytic side, $\ell^1(G)$ is the Banach algebra of absolutely summable complex-valued functions on $G$ with convolution product; it embeds continuously into the reduced group $C^*$-algebra $C^*_r(G)$, the completion of the complex group ring $\mathbb{C}G$ for its action by left convolution on the Hilbert space $\ell^2(G)$. Topological K-theory $K^{\mathrm{top}}_*(\ell^1(G))$ is defined as for Banach algebras generally: $K_0$ is the Grothendieck group of finitely generated projective $\ell^1(G)$-modules, and the higher groups are built from homotopy classes of invertible matrices over $\ell^1(G)$ and over its suspensions. The Baum–Connes conjecture with trivial coefficients predicts that an assembly map from the equivariant K-homology $K^G_*(\underline{E}G)$ — where $\underline{E}G$ denotes the classifying space for proper $G$-actions, a proper $G$-CW-complex universal among such spaces — to $K_*(C^*_r(G))$ is an isomorphism; the Bost conjecture is the variant of this statement with the Banach algebra $\ell^1(G)$ in place of the reduced $C^*$-algebra. The Baum–Connes assembly map factors through the Bost assembly map: it is the composite of $\mu^G$ with the map on K-theory induced by the inclusion $\ell^1(G)\hookrightarrow C^*_r(G)$ (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 $\ell^1(G)$ is closed under holomorphic functional calculus in $C^*_r(G)$, by the classical density theorem for such dense subalgebras (Swan, Karoubi, Bost), but it is itself open in general: $\ell^1(G)$ fails to be stable under holomorphic calculus already when $G$ is a non-compact semisimple Lie group, and the Bost conjecture is known for lattices in reductive Lie groups, such as $\mathrm{SL}_3(\mathbb{Z})$, for which the Baum–Connes conjecture remains open.

The $\ell^1$-formulation is more flexible than the $C^*$-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 $C^*$-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 $\mu^G\colon K^G_*(\underline{E}G)\longrightarrow K^{\mathrm{top}}_*(\ell^1(G))$ an isomorphism for every countable discrete group $G$? Here $\underline{E}G$ is the classifying space for proper $G$-actions, $K^G_*(\underline{E}G)$ its equivariant K-homology, $\ell^1(G)$ the convolution Banach algebra of absolutely summable functions on $G$, and $K^{\mathrm{top}}_*(\ell^1(G))$ the topological K-theory of that Banach algebra. The task is to resolve this question: either prove that $\mu^G$ is an isomorphism for every countable discrete group $G$, or exhibit a countable discrete group $G$ for which $\mu^G$ 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 $\mathbb{Z}G$-modules, for every group $G$.

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
3. M. P. Gómez Aparicio, P. Julg, A. Valette, "The Baum–Connes conjecture: an extended survey", in: Advances in Noncommutative Geometry, Springer, 2019, pp. 127–244, DOI 10.1007/978-3-030-29597-4_3, https://doi.org/10.1007/978-3-030-29597-4_3 (also arXiv:1905.10081, https://arxiv.org/abs/1905.10081)
4. N. Higson, G. Kasparov, "E-theory and KK-theory for groups which act properly and isometrically on Hilbert space", Inventiones mathematicae 144 (2001), 23–74, DOI 10.1007/s002220000118, https://doi.org/10.1007/s002220000118
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)
7. W. Paravicini, "The Bost conjecture and proper Banach algebras", Journal of Noncommutative Geometry 7 (2013), 191–202, DOI 10.4171/jncg/113, https://doi.org/10.4171/jncg/113
8. J. Wang, Z. Xie, G. Yu, "$\ell^1$-higher index, $\ell^1$-higher rho invariant and cyclic cohomology", Proceedings of Symposia in Pure Mathematics 105 (Cyclic Cohomology at 40: Achievements and Future Prospects), American Mathematical Society, 2023, pp. 535–579, DOI 10.1090/pspum/105/25, https://doi.org/10.1090/pspum/105/25 (also arXiv:2206.09913, https://arxiv.org/abs/2206.09913)