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Survey on the Farrell-Jones Conjecture

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Pith's one-line read This survey claims the Full Farrell-Jones Conjecture is now known for a very broad, robustly closed class of groups, and it implies many classical K- and L-theoretic conjectures for all of them.

desk verdict A careful, useful survey of the Farrell-Jones landscape that leans heavily on the author's forthcoming book, with one citation gap to check around directed colimit inheritance. read the letter →

arxiv 2507.11337 v1 pith:FNC7RNBE submitted 2025-07-15 math.KT math.GRmath.GT

Wolfgang Lueck This is my paper · ORCID
classification math.KTmath.GRmath.GT MSC 19-0219A3119B2819G24
keywords Farrell-JonesConjecturealgebraicK-theoryL-theorygroupringsassemblymapsvirtuallycyclicsubgroupshyperbolicgroupsdirectedcolimits
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

This survey presents the Farrell-Jones Conjecture on the algebraic K- and L-theory of group rings and argues that it is now established for a very large class of groups. The class $\mathbf{FJ}$ of groups satisfying the Full Farrell-Jones Conjecture contains hyperbolic groups, CAT(0)-groups (groups acting properly and cocompactly on CAT(0) spaces), virtually solvable groups, lattices, S-arithmetic groups, mapping class groups, braid groups, Coxeter groups, and fundamental groups of 3-manifolds, and it is closed under subgroups, directed colimits, free products, and graph products. If a group lies in $\mathbf{FJ}$, every variant of the conjecture in the literature holds for it, delivering a long list of classical consequences: vanishing of Whitehead and projective class groups for torsionfree groups, the Borel Conjecture, and the Bass, Novikov, and Serre conjectures. The survey's status theorem, Theorem 8.12, is the load-bearing result, and its heredity under directed colimits lets the conjecture pass to exotic groups and reduces the problem for all groups to one universal finitely presented group.

What carries the argument

The object that carries the argument is the assembly map for a family of subgroups, specialized to the family $\mathcal{VCY}$ of virtually cyclic subgroups. For a group $G$, the K-theoretic conjecture asks that the map $H^G_n(E_{\mathcal{VCY}}(G); \mathbf{K}_{\mathcal{A}}) \to K_n(\mathcal{A}[G])$ induced by the projection to $G/G$ be bijective for every $n$, where $\mathcal{A}$ is an additive $G$-category; the L-theoretic version is the analogous statement for $L^{\langle -\infty\rangle}$. The Full conjecture applies this to the wreath products $G \wr F$ for every finite group $F$, which is the formulation that makes the class $\mathbf{FJ}$ robust under the inheritance properties listed in Theorem 8.12. The proof strategies surveyed—assembly maps, controlled topology, flow spaces, and transfers—are the machinery that established $\mathbf{FJ}$ for the listed groups.

What would settle it

Look at the cited proof of the directed-colimit rule. A concrete test: take a directed colimit of hyperbolic groups, such as a group with coarsely embedded expanders, and compute whether the K-theoretic assembly map $H^G_n(E_{\mathcal{VCY}}(G); \mathbf{K}_{\mathcal{A}}) \to K_n(\mathcal{A}[G])$ is bijective for all $n$ and all additive $G$-categories; a failure would refute Theorem 8.12(ii)(f) and, with it, the claim that the Full conjecture is open only in the listed families.

Watch

Extended reading notes

Core claim

The central claim of the paper is Theorem 8.12: the class $\mathbf{FJ}$ of Farrell-Jones groups is both broad and closed under many constructions. It contains hyperbolic groups, finite-dimensional CAT(0)-groups, virtually solvable groups, (not necessarily cocompact) lattices in path-connected second-countable locally compact Hausdorff groups, fundamental groups of connected manifolds of dimension at most three, $\mathrm{GL}_n(\mathbb{Q})$ and $\mathrm{GL}_n$ over function fields, S-arithmetic groups, mapping class groups, braid groups, Coxeter groups, and fundamental groups of graphs of abelian or virtually cyclic groups. The class is closed under subgroups, finite products, certain group extensions, directed colimits with arbitrary structure maps, free products, overgroups of finite index, and graph products. Since the Full Farrell-Jones Conjecture implies all K-theoretic and L-theoretic versions—including the coefficient versions for additive and higher categories—every group in $\mathbf{FJ}$ inherits the full set of applications outlined in the survey. The paper also notes that no group is known to violate the conjecture, that the open cases include Thompson's groups, $\mathrm{Out}(F_n)$ for $n\ge 3$, Artin groups, and linear groups, and that a positive answer for one universal finitely presented group would settle the conjecture for all groups.

Load-bearing premise

The whole breadth of the result rests on one inheritance rule: a group built as a limit of a directed system of Farrell-Jones groups, with arbitrary connecting maps, is again Farrell-Jones, and the survey cites the proof of this rule to the literature rather than reproducing it.

Editorial extensions

If this is right

  • Every torsionfree group in $\mathbf{FJ}$ has vanishing Whitehead group, vanishing reduced projective class group, and vanishing negative $K$-groups, so the finiteness obstruction and the $s$-cobordism theorem apply without hidden torsion.
  • The Borel Conjecture—topological rigidity of aspherical closed manifolds—follows in dimension at least 5 for every torsionfree fundamental group in $\mathbf{FJ}$.
  • The Full conjecture implies the Bass, Borel, Novikov, and Serre conjectures for each Farrell-Jones group, as well as the stable Cannon Conjecture and the product decomposition theorem for aspherical manifolds.
  • Because $\mathbf{FJ}$ is closed under directed colimits with arbitrary structure maps, every directed colimit of hyperbolic groups, including groups with coarsely embedded expanders, satisfies the Full conjecture.
  • If one universal finitely presented group satisfies the Full conjecture, then every group satisfies it.

Reading between the lines

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

  • Editorial inference: the universal-finitely-presented reduction points toward an algorithmic route—verify the Full conjecture for a single, explicitly constructed group—but the paper stops short of suggesting this is practical.
  • Editorial inference: because arbitrary directed colimits are allowed, any future counterexample would have to be a group that cannot be written as such a limit of Farrell-Jones groups, which sharply narrows the search space implied by the survey.
  • Editorial inference: if the status report is accurate, the groups still open—$\mathrm{Out}(F_n)$, Artin groups, Thompson's groups, linear groups—differ from the known ones less by geometry than by the present reach of flow-space and transfer techniques.
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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 / 7 minor

Summary. This survey gives a broad account of the Farrell-Jones Conjecture in algebraic K- and L-theory of group rings. It first introduces the classical torsionfree-group consequences (projective class groups, Whitehead groups, lower and higher K-theory, L-theory) and their applications to the Borel Conjecture, aspherical manifolds, the stable Cannon Conjecture, automorphism groups, and Poincaré duality groups. It then formulates the Full Farrell-Jones Conjecture 8.10 with coefficients in additive categories and in right exact G-∞-categories, defines the class FJ of Farrell-Jones groups as those satisfying it, and states Theorem 8.12 asserting that FJ contains many geometric and algebraic classes of groups and is closed under several operations, including directed colimits with arbitrary structure maps. The last sections sketch proof methods and propose a conditional reduction of the conjecture for all groups to a single universal finitely presented group.

Significance. If the statements collected here are accurate, the survey is a valuable and accessible reference for a central conjecture in algebraic and geometric topology. Its main strength is that it packages a large body of published theorems into a unified framework: the class FJ of Farrell-Jones groups, the asserted inheritance properties, and the implication from the Full Farrell-Jones Conjecture to numerous applications. The paper is also unusually candid about a known gap in the literature, acknowledging in Remark 7.31 that a theorem from [17] required an additional hypothesis and adjusting Theorem 7.30 accordingly. The breadth of the claimed class FJ and the conditional all-groups reduction in Remark 11.4 are substantial, but they rest on Theorem 8.12(ii)(f), whose higher-categorical version needs explicit verification.

major comments (3)
  1. [§8.8, Theorem 8.12(ii)(f)] The closure of FJ under directed colimits with arbitrary structure maps is load-bearing: it is used to place groups with expanders inside FJ and to reduce the conjecture for all groups to a single finitely presented group in Remark 11.4. However, the proof is not supplied. The theorem is justified by a blanket reference to [69, Section 16.2], an unpublished book by the author, while the published paper [5] cited in the bibliography predates Conjecture 8.9, which concerns K-theory with coefficients in right exact G-∞-categories and is part of Definition 8.11 via Conjecture 8.10. As cited, [5] establishes colimit closure for the additive-category versions 8.7 and 8.8. Since Definition 8.11 requires Conjecture 8.9 for the wreath products, the cited support does not on its face establish that a directed colimit of Farrell-Jones groups is a Farrell-Jones group. Please provide a proof or a published reference covering the higher-categorical clause, or adjust the definition of FJ and the statements that depend on this closure.
  2. [§8.7] The assertion that the Full Farrell-Jones Conjecture 8.10 implies all the variants listed in Sections 2 through 7, including the higher-categorical Conjecture 8.9 and 'fibered versions', is a central organizing claim of the survey. The reader is referred only to [69, Section 13.11], again an unpublished book. Since this implication is what makes the class FJ relevant to the applications in Sections 2–7, the paper should either prove the implications or cite published references that establish them.
  3. [Remark 11.4] The conditional reduction to a single universal finitely presented group is stated as an 'amusing observation' but is in fact a strong theorem with several nontrivial steps: a group is a directed colimit of its finitely generated subgroups, a finitely generated group is a directed colimit of finitely presented groups, and every finitely presented group embeds in the universal group U. The first two steps use Theorem 8.12(ii)(f) with arbitrary structure maps, not just inclusions. Since that closure property is the point at issue in the first major comment, Remark 11.4 should state this dependence explicitly and should not present the reduction as fully established until the higher-categorical case of Theorem 8.12(ii)(f) is justified.
minor comments (7)
  1. [§7.4] The phrase 'Virtually Fibering Conjecture' should be 'Virtual Fibering Conjecture' or 'Virtually Fibered Conjecture'.
  2. [§5.1] The text attributes a construction of the non-connective K-theory spectrum to 'Schlichting [23]', but reference [23] is Cárdenas–Pedersen; a reference to Schlichting's work appears to be missing from the bibliography.
  3. [Theorem 7.16] The symbol cM is used both for the G-covering of M and for the compact topological manifold whose interior is homeomorphic to that covering; please use different notation for these two objects.
  4. [Remark 7.31] The sentence 'Theorems 7.12 and 7.16 remain true with adding any further hypothesis' should read 'without adding any further hypothesis'.
  5. [§8.7] In the list of implications, '5.3 6.10' should be '5.3 and 6.10'.
  6. [§10.4] The symbol p∗ is used both for the transfer map Wh(π1(M)) → Wh(π1(ST M)) and for the induced pushforward Wh(π1(ST M)) → Wh(π1(M)); please disambiguate the notation.
  7. [References] Reference [19], the erratum to [17], is listed with the same volume and page numbers as [17] (Ann. of Math. 143(3):435–467); the bibliographic data should be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the survey reports external theorems; its self-citations point to real prior work and no claim reduces by construction to its own inputs.

full rationale

This paper is a survey, not a derivation from first principles within the manuscript. Its central content, Theorem 8.12, is explicitly attributed to [69, Section 16.2] and to a list of published papers ([4,5,6,7,8,9,21,40,41,44,57,94,95]); the survey does not claim to prove these results itself. Definitions such as the class FJ (Definition 8.11) are stated in terms of the Full Farrell-Jones Conjecture (Conjecture 8.10), and the later assertions about which groups lie in FJ are imported from the cited literature, not derived from the definition alone. The applications in Sections 7 and 9 are stated as conditional implications, e.g., Theorem 7.5 derives the Borel Conjecture from Conjectures 4.12 and 6.10; this is a genuine implication, not a circular reduction. The self-references to [69] (the author's book) and to [5] (Bartels–Echterhoff–Lück) are load-bearing for the breadth of FJ, but they are citations to external, checkable results with assumptions independent of the present survey; under the stated rules, such citations count as real evidence and do not raise the circularity score. A skeptical reader might worry that [69, Section 16.2] may prove the directed-colimit inheritance only for the additive-category versions (Conjectures 8.7 and 8.8) and not for the higher-∞-category version (Conjecture 8.9) used in Definition 8.11; that is a verification question about the cited literature, not a circularity within this paper. The paper also honestly records limitations, such as the added TOP-reduction hypothesis in Theorem 7.30 due to an erratum in the literature, further indicating that the survey is not masking a circular dependency. No equation or fitted parameter is renamed as a prediction, and no uniqueness claim is imported solely from the author's prior work. Therefore no significant circularity is present.

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

No fitted constants or invented entities appear. The survey rests on the standard formalism of K- and L-theory spectra, classifying spaces for families, and the surgery exact sequence, plus the cited inheritance theorems that make the class FJ large. The most load-bearing external input is the directed colimit closure, which is used to extend FJ to exotic groups and to the universal-group reduction.

assumptions (4)
  • domain assumption The Full Farrell-Jones Conjecture (Conjecture 8.10) implies all variant conjectures discussed in the paper.
    Used throughout Section 8.7 and Section 9 to transfer FJ to applications; the proof is referenced to [69, Section 13.11], not given in the paper.
  • domain assumption FJ is closed under directed colimits with arbitrary structure maps (Theorem 8.12(ii)(f)).
    Underlies the claim that colimits of hyperbolic groups satisfy FJ and the reduction in Remark 11.4; cited to Bartels-Echterhoff-Luck and the author's book.
  • standard math The non-connective K-theory and L-theory spectra with the stated homotopy groups exist (Sections 5.1 and 6.2).
    Background formalism invoked without proof; standard in the field and cited to Pedersen-Weibel, Schlichting, and Luck-Steimle.
  • domain assumption The Surgery Exact Sequence (7.8) and the Atiyah-Hirzebruch spectral sequence comparison in the proof sketch of Theorem 7.5 are valid.
    Used to derive the Borel Conjecture from FJ; the paper gives only a sketch and refers to [70, Chapter 19] for details.

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Cite this review

Pith. "Pith review of Survey on the Farrell-Jones Conjecture." pith.science (2026). https://pith.science/paper/FNC7RNBE

@misc{pith2026250711337,
  author       = {Pith},
  title        = {Pith review of: Survey on the Farrell-Jones Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNC7RNBE}},
  note         = {Machine review of arXiv:2507.11337}
}
read the original abstract

This is a survey on the Farrell-Jones Conjecture about the algebraic K- and L-theory of groups rings and its applications to algebra, geometry, group theory, and topology.

Discussion (0). Continue with ORCID to comment.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Some remarks on $h$-cobordisms between smooth 4-manifolds

    math.GT 2026-04 unverdicted novelty 6.0 of 10

    New sufficient conditions make every Whitehead torsion realizable by inertial smooth h-cobordisms in dimension 4, and Cohen's dim invariant is shown to obstruct the standard construction.

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