Burghelea's 1985 computation decomposes the periodic cyclic homology of the rational group algebra QG\mathbb{Q}G over the conjugacy classes of GG; the contribution of the infinite-order classes is a direct sum of groups T∗(x,Q)T_*(x,\mathbb{Q}) built from the inverse systems of the rational homology of the reduced centralizers Nx=Gx/⟨x⟩N_x=G_x/\langle x\rangle, bonded by Gysin maps. Burghelea conjectured that T∗(x,Q)=0T_*(x,\mathbb{Q})=0 for every infinite-order conjugacy class whenever K(G,1)K(G,1) has the homotopy type of a finite CW-complex, which would complete this computation and implies the Bass conjecture. The conjecture is proved for many classes of groups (hyperbolic, arithmetic, CAT(0) of finite rational homological dimension, systolic, mapping class groups, 3-manifold groups, and more), but it remains open in general: all known counterexamples live among groups admitting no finite classifying space, and no proof — and no genuine counterexample — is known for the finite-K(G,1)K(G,1) class.

Background

Hochschild homology HH∗(A)HH_*(A) and cyclic homology HC∗(A)HC_*(A) are homology theories for an algebra AA over a commutative ring, introduced in the early 1980s. They are connected by the Connes exact sequence, whose periodicity operator S:HCn(A)→HCn−2(A)S:HC_n(A)\to HC_{n-2}(A) yields, on passing to an inverse limit, the periodic cyclic homology PHC∗(A)PHC_*(A), ∗=0,1*=0,1. A classifying space K(G,1)K(G,1) of a discrete group GG is a connected CW-complex with fundamental group GG and contractible universal cover.

For the rational group algebra QG\mathbb{Q}G, Burghelea computed these theories explicitly. Writing ⟨G⟩\langle G\rangle for the set of conjugacy classes of GG, ⟨G⟩fin\langle G\rangle_{\mathrm{fin}} and ⟨G⟩∞\langle G\rangle_\infty for the classes of finite- and infinite-order elements, GxG_x for the centralizer of xx, and Nx=Gx/⟨x⟩N_x=G_x/\langle x\rangle for the reduced centralizer, one has HHn(QG)≅⨁[x]∈⟨G⟩Hn(Gx;Q)HH_n(\mathbb{Q}G)\cong\bigoplus_{[x]\in\langle G\rangle}H_n(G_x;\mathbb{Q}), together with a compatible decomposition of HCnHC_n and of PHC∗PHC_*; the infinite-order part of PHC∗(QG)PHC_*(\mathbb{Q}G) is the direct sum of groups T∗(x,Q)T_*(x,\mathbb{Q}), each defined by the short exact sequence

0→lim⁡1{H∗−1+2n(Nx;Q)}→T∗(x,Q)→lim⁡{H∗+2n(Nx;Q)}→0,0\to\lim^1\{H_{*-1+2n}(N_x;\mathbb{Q})\}\to T_*(x,\mathbb{Q})\to\lim\{H_{*+2n}(N_x;\mathbb{Q})\}\to 0,

where the bonding maps are the Gysin homomorphisms of the circle fibration B⟨x⟩≃S1→BGx→BNxB\langle x\rangle\simeq S^1\to BG_x\to BN_x. Here lim⁡\lim is the inverse limit over nn and lim⁡1\lim^1 its first derived functor; the inverse system is Mittag–Leffler when the images of the transition maps stabilize in each degree, and Mittag–Leffler forces lim⁡1=0\lim^1=0, so T∗(x,Q)=0T_*(x,\mathbb{Q})=0 exactly when the limit and derived limit both vanish.

At the end of his paper Burghelea conjectured that if K(G,1)K(G,1) has the homotopy type of a finite CW-complex, then T∗(x,Q)=0T_*(x,\mathbb{Q})=0 for all [x]∈⟨G⟩∞[x]\in\langle G\rangle_\infty; this would complete his computation for such groups. There is also a KK-theoretic stake: the Hattori–Stallings rank of idempotent matrices over QG\mathbb{Q}G (the map recording the coefficients of a trace per conjugacy class) factors through the groups T0(x,Q)T_0(x,\mathbb{Q}), so the conjecture implies the strong Bass conjecture — vanishing of the infinite-order components of that rank — and hence, for torsion-free GG, the idempotent conjecture that e2=ee^2=e in QG\mathbb{Q}G has only the solutions e=0,1e=0,1. It also fits into the rationalized algebraic Baum–Connes assembly picture.

Known cases accumulate: Burghelea himself handled fundamental groups of Riemannian manifolds of nonpositive sectional curvature; Eckmann proved the conjecture for nilpotent and torsion-free solvable groups of finite rational homological dimension, for linear groups over fields of characteristic zero of finite rational homological dimension, and for groups of rational cohomological dimension at most 22; Ji proved it for hyperbolic groups, arithmetic groups, and groups of polynomial growth. Dranishnikov proved it for all groups of finite integral cohomological dimension whose reduced centralizers NxN_x each have trivial rational homology in infinitely many degrees. Engel and Marcinkowski proved it for CAT(0) groups of finite rational homological dimension (CAT(0) means geodesic metric spaces with nonpositively curved triangles in the comparison sense, on which the group acts properly and cocompactly; this class includes Coxeter and right-angled Artin groups), for systolic groups (groups acting properly and cocompactly on systolic, i.e. flag-no-square simplicial complexes), for mapping class groups of surfaces, for discrete subgroups of almost connected Lie groups, and for fundamental groups of 3-manifolds; the property is also inherited by relatively hyperbolic supergroups (relatively hyperbolic generalizing knot-group-type hyperbolicity relative to a peripheral subgroup). Huang and Osajda deduced it for large-type Artin groups via systolicity. Engel and Marcinkowski further proposed two conjectures for groups of finite asymptotic dimension (a coarse dimension invariant of the group as a metric space): finite rational homological dimension, and finite asymptotic dimension of all quotients Cg/⟨g⟩C_g/\langle g\rangle; both together imply the Burghelea conjecture for such groups and are verified for most known classes.

The conjecture cannot simply hold for all groups: Burghelea constructed a countable, non-finitely generated counterexample to the unrestricted vanishing statement; Dranishnikov and Hull constructed a finitely generated counterexample via Osin's small-cancellation theory over relatively hyperbolic groups; Engel and Marcinkowski constructed a finitely presented counterexample of type F∞F_\infty (a central extension 1→Z→G→F→11\to\mathbb{Z}\to G\to F\to 1 of Thompson's group FF of piecewise-linear homeomorphisms of [0,1][0,1]) with T∗(x,Q)≅Q×QT_*(x,\mathbb{Q})\cong\mathbb{Q}\times\mathbb{Q}, as well as a finitely generated counterexample with finite decomposition complexity (a coarse invariant weaker than finite asymptotic dimension). Crucially, none of these counterexamples admits a finite classifying space: the Thompson's FF extension is of type F∞F_\infty, meaning K(G,1)K(G,1) is a CW-complex with finitely many cells in each dimension, but its integral cohomology contains torsion in infinitely many degrees, so its cohomological dimension is infinite. Dranishnikov and Hull state explicitly (2019 revision) that no counterexample to the original finite-K(G,1)K(G,1) version is known.

Problem Statement

The original Burghelea conjecture. Prove or disprove: if the classifying space K(G,1)K(G,1) of a discrete group GG has the homotopy type of a finite CW-complex, then T∗(x,Q)=0T_*(x,\mathbb{Q})=0 for every conjugacy class [x]∈⟨G⟩∞[x]\in\langle G\rangle_\infty of infinite-order elements and both parities ∗=0,1*=0,1 — equivalently, for every such GG and xx, the inverse system {H∗+2n(Nx;Q),S}\{H_{*+2n}(N_x;\mathbb{Q}),S\} of the rational homology of the reduced centralizer Nx=Gx/⟨x⟩N_x=G_x/\langle x\rangle, bonded by the Gysin maps of the fibration S1→BGx→BNxS^1\to BG_x\to BN_x, satisfies the Mittag–Leffler condition with vanishing lim⁡\lim and vanishing lim⁡1\lim^1. The task is this headline claim; the two sub-questions the source literature embeds in it, recorded here as context of the headline claim rather than as separately posed questions in their own right, are: (i) is the centralizer-homology hypothesis of Dranishnikov's theorem — trivial rational homology of every NxN_x in infinitely many degrees — automatic, or does finiteness of K(G,1)K(G,1) (or even just finite integral cohomological dimension) already force the vanishing; and (ii) which groups, more generally, have all their Gysin systems Mittag–Leffler with vanishing limits. A refutation must exhibit a group whose K(G,1)K(G,1) is itself homotopy equivalent to a finite CW-complex: the known counterexamples (Burghelea's countable group, the Dranishnikov–Hull finitely generated group, the Engel–Marcinkowski finitely presented type-F∞F_\infty extension of Thompson's group FF) all lie outside this class and do not settle the question. A resolution would also advance the strong Bass and idempotent conjectures for the class of groups involved.

References

  1. D. Burghelea, The cyclic homology of the group rings, Commentarii Mathematici Helvetici 60 (1985) 354–365, DOI: 10.1007/BF02567420, https://doi.org/10.1007/BF02567420
  2. D. Burghelea, Erratum to “The cyclic homology of the group rings” (corrects Propositions II and IIp of the 1985 paper), Commentarii Mathematici Helvetici, 2023, DOI: 10.4171/cmh/561, https://doi.org/10.4171/cmh/561
  3. A. Dranishnikov, On Burghelea Conjecture, arXiv:1612.08700 (2016, revised through 2021), https://arxiv.org/abs/1612.08700
  4. Alexander Engel, Michał Marcinkowski, Burghelea conjecture and asymptotic dimension of groups, arXiv:1610.10076 (2016, v3 2018); published in the Journal of Topology and Analysis 12, 321–356 (online 2018), DOI: 10.1142/S1793525319500559, https://doi.org/10.1142/S1793525319500559
  5. A. Dranishnikov, M. Hull, A finitely generated group that does not satisfy the generalized Burghelea Conjecture, arXiv:1701.03165 (2017, v2 2019), https://arxiv.org/abs/1701.03165
  6. B. Eckmann, Cyclic homology of groups and the Bass conjecture, Commentarii Mathematici Helvetici 61 (1986) 193–202, DOI: 10.1007/BF02621911, https://doi.org/10.1007/BF02621911
  7. R. Ji, Nilpotency of Connes' periodicity operator and the idempotent conjectures, K-Theory 9 (1995) 59–76, DOI: 10.1007/BF00965459, https://doi.org/10.1007/BF00965459
  8. Jingyin Huang, Damian Osajda, Large-type Artin groups are systolic, Proceedings of the London Mathematical Society 120, 95–123 (online 2019), DOI: 10.1112/plms.12284, arXiv:1706.05473 (2017), https://doi.org/10.1112/plms.12284