{"id":"a8bf901c-c548-4408-a473-e5ca5fdc7eb0","arxiv_id":"2507.11337","paper_version":1,"verdict":"UNVERDICTED","confidence":"MODERATE","novelty_score":0.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The Farrell-Jones Conjecture is known for hyperbolic, CAT(0), arithmetic, and many other groups, and it implies a long list of conjectures about group rings and aspherical manifolds.","lead":"This paper surveys the Farrell-Jones Conjecture, a central open problem about the algebraic K- and L-theory of group rings. It lays out which groups are known to satisfy the conjecture and what the conjecture implies for manifolds, group theory, and topology.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing point is the directed-colimit closure of FJ, but the cited support may not cover the higher-categorical clause in the survey's definition of FJ; verify [69, §16.2] before relying on Remark 11.4.","rationale":"The reader correctly identifies Theorem 8.12(ii)(f), closure under directed colimits with arbitrary structure maps, as the most load-bearing assumption: it is used to place exotic groups built as colimits of hyperbolic groups into FJ and to reduce the Full FJ for all groups to a single universal finitely presented group in Remark 11.4. My stress-test agrees that this is where the central claim is least secure. However, I do not find a known counterexample or an internal inconsistency. The real issue is a verification gap in the support chain: the survey's Definition 8.11 makes FJ depend on Conjecture 8.9, the K-theoretic Farrell-Jones conjecture with coefficients in right exact G-∞-categories. The classical inheritance theorem of Bartels–Echterhoff–Lück, which the reader cites, was published in 2008 and concerns additive-category coefficients; it does not obviously cover the higher-categorical version. The survey's blanket reference to the author's upcoming book [69, Section 16.2] may well contain a proof, but that proof is not available to a reader of the arXiv preprint and is not independently verifiable from the text. This is exactly the kind of missing-support point the review should flag. If the book does prove the colimit closure for Conjecture 8.9, the concern dissolves and the survey's Theorem 8.12 is accurate. If it does not, then the definition of FJ used in Theorem 8.12(ii)(f) is stronger than the cited results establish, and the expander examples and the universal-group reduction require either a weaker notion of FJ or a new proof. The concrete test I propose settles this by checking the exact quantification in [69, Section 16.2] and, independently, by locating a peer-reviewed proof for the higher-categorical case. Since this is a verification gap rather than a demonstrated error, the appropriate verdict remains UNVERDICTED, matching the reader's assessment.","tokens_in":33871,"tokens_out":13980,"duration_ms":171033,"concrete_test":"Obtain [69, Section 16.2] (or the corresponding result in the book manuscript) and read the precise statement of the directed-colimit inheritance theorem. Check whether the theorem quantifies over all right exact G-∞-categories (Conjecture 8.9) or only over additive G-categories with involution (Conjectures 8.7/8.8). Independently, search for a peer-reviewed proof that the class of groups satisfying the K-theoretic Farrell-Jones conjecture with higher-categorical coefficients is closed under directed colimits with arbitrary structure maps. If no such proof exists, Theorem 8.12(ii)(f) should be restricted to the additive-category version of the Full FJ, and Remark 11.4 should be flagged accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 8.12(ii)(f) — closure of FJ under directed colimits with arbitrary structure maps — is the load-bearing assumption. It is the only route by which groups with expanders enter FJ and it underwrites the conditional all-groups reduction in Remark 11.4. The survey defines FJ via the Full Conjecture 8.10, which includes the K-theoretic Farrell-Jones conjecture with coefficients in right exact G-∞-categories (Conjecture 8.9). The inheritance theorem of Bartels–Echterhoff–Lück [5] cited in the paper predates this higher-categorical formulation and, as cited, establishes colimit closure only for the additive-category versions (8.7/8.8). Theorem 8.12 is justified by a blanket reference to [69, Section 16.2], an unpublished book by the same author. If [69] proves the colimit closure only for 8.7/8.8 and not for 8.9, then the class FJ of Definition 8.11 is stronger than what the cited proofs establish, and both the expander examples and Remark 11.4 lose their support. This is a verification gap, not a demonstrated counterexample.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34036,"tokens_out":9980,"duration_ms":116441,"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":[{"comment":"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.","section":"§8.8, Theorem 8.12(ii)(f)"},{"comment":"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.","section":"§8.7"},{"comment":"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.","section":"Remark 11.4"}],"minor_comments":[{"comment":"The phrase 'Virtually Fibering Conjecture' should be 'Virtual Fibering Conjecture' or 'Virtually Fibered Conjecture'.","section":"§7.4"},{"comment":"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.","section":"§5.1"},{"comment":"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.","section":"Theorem 7.16"},{"comment":"The sentence 'Theorems 7.12 and 7.16 remain true with adding any further hypothesis' should read 'without adding any further hypothesis'.","section":"Remark 7.31"},{"comment":"In the list of implications, '5.3 6.10' should be '5.3 and 6.10'.","section":"§8.7"},{"comment":"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.","section":"§10.4"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a survey by the leading expert in the area, and the mathematical core is likely sound. However, the load-bearing statements about the class FJ are justified by reference to the author's unpublished book [69], and the published paper [5] cited for colimit closure appears to predate the higher-categorical formulation in Conjecture 8.9. For a journal publication, this reliance on an unavailable source is risky: a referee cannot verify Theorem 8.12(ii)(f) or the implication claims in §8.7. I would recommend asking the author to supply published references or an appendix with proofs for these points before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sure, here's my take. This is a survey, explicitly an extract of Lück's forthcoming book [69]. That is the honest framing, and it is what the paper is: a well-organized map of the Farrell-Jones conjecture, its variants, applications, and proof strategies. No new theorems. The value is organizational: one place to look up which groups are known to satisfy the full conjecture and which applications follow. It also does something I like: it tells you when a statement has a known erratum. Theorem 7.30 carries an extra TOP-reduction assumption because of [17], and the paper says so, pointing to the erratum [19]. That kind of transparency is exactly what a survey should do.\n\nThe real soft spot is Theorem 8.12(ii)(f), closure of FJ under directed colimits with arbitrary structure maps. This is load-bearing. It is how groups with expanders enter FJ, and Remark 11.4 uses it to reduce the whole conjecture to a single universal finitely presented group. The proof is not reproduced; you get a blanket reference to [69, Section 16.2]. The paper also cites Bartels-Echterhoff-Lück [5] for inheritance under colimits. But the definition of FJ here is the Full Conjecture 8.10, which includes Conjecture 8.9 — K-theory with coefficients in right exact G-∞-categories. My understanding is that [5] establishes the colimit inheritance for the additive-category versions 8.7/8.8, and it is not clear that the ∞-categorical version was covered there. So there is a real verification gap: either [69, §16.2] proves 8.9's colimit closure, in which case fine, or it does not, and then both the expander examples and Remark 11.4 lose their cited support. This is not a demonstrated counterexample, but it is a gap that should be fixed before the survey appears.\n\nOther minor issues: a typo in 'Virtually Fibering Conjecture' in the text, and the prose is occasionally compressed to the point of being hard to follow for a newcomer. None of that matters much.\n\nWho is this for? People who want an up-to-date orientation in K- and L-theoretic assembly conjectures, especially graduate students or researchers entering the area. It deserves a serious referee: a survey by a leading expert, with real organizational value, and one specific citation gap that a referee can check. I'd send it to review, with instructions to verify the colimit inheritance against [69].","headline":"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.","tokens_in":34582,"tokens_out":1912,"would_cite":true,"duration_ms":21736,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19-02","19A31","19B28","19G24"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Farrell-Jones Conjecture","algebraic K-theory","algebraic L-theory","group rings","assembly maps","virtually cyclic subgroups","hyperbolic groups","directed colimits"],"falsifier":"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.","tokens_in":33606,"feed_emoji":"","tokens_out":11371,"duration_ms":119496,"temperature":0.7,"pith_summary":"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.","feed_headline":"One conjecture covers hyperbolic, arithmetic, and 3-manifold groups","feed_subtitle":"If the Full conjecture holds, Borel, Bass, Novikov, and Serre follow for all these groups.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The book from which the survey is extracted; it supplies the proofs and references for the status theorem and its inheritance properties.","marker":"[69]"},{"why":"Gives the original formulation of the Farrell-Jones Conjecture with rings as coefficients, the statement the survey presents and extends.","marker":"[44]"},{"why":"Supplies the proof that the class of Farrell-Jones groups is closed under directed colimits, the load-bearing inheritance rule.","marker":"[5]"},{"why":"Establishes the K-theoretic Farrell-Jones Conjecture for hyperbolic groups, a central input to the list.","marker":"[7]"},{"why":"Establishes the L-theoretic Farrell-Jones Conjecture for hyperbolic groups, used in the geometric applications.","marker":"[8]"},{"why":"Proves the Farrell-Jones Conjecture for mapping class groups, one of the classes in Theorem 8.12.","marker":"[4]"},{"why":"Proves the conjecture for cocompact lattices in virtually connected Lie groups, supporting the lattice entries.","marker":"[6]"},{"why":"Proves K- and L-theory results for $\\mathrm{GL}_n(\\mathbb{Z})$, supporting the S-arithmetic group entry.","marker":"[9]"},{"why":"Constructs a universal finitely presented group, used in the reduction of the conjecture for all groups to one group.","marker":"[55]"},{"why":"Proves rational injectivity of the K-theoretic assembly map for every group, the all-groups statement in Section 11.","marker":"[103]"}],"fun_headline_variants":["Farrell-Jones conjecture: one theorem for many group families","Survey: FJ class broad, no counterexamples known yet","If true, FJ implies Borel, Bass, Novikov, Serre conjectures","Farrell-Jones: from hyperbolic to solvable, all in one class","Open cases remain: Thompson, Out(F_n), Artin groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Farrell-Jones conjecture: one theorem for many group families","Survey: FJ class broad, no counterexamples known yet","If true, FJ implies Borel, Bass, Novikov, Serre conjectures","Farrell-Jones: from hyperbolic to solvable, all in one class","Open cases remain: Thompson, Out(F_n), Artin groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000536,"raw_usage":{"total_tokens":2519,"prompt_tokens":836,"completion_tokens":1683,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":1589}},"tokens_in":452,"tokens_out":1683,"duration_ms":16407,"temperature":1.0,"reasoning_tokens":1589,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:10:56.151022+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The book from which the survey is extracted; it supplies the proofs and references for the status theorem and its inheritance properties."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the original formulation of the Farrell-Jones Conjecture with rings as coefficients, the statement the survey presents and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs a universal finitely presented group, used in the reduction of the conjecture for all groups to one group."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves rational injectivity of the K-theoretic assembly map for every group, the all-groups statement in Section 11."}],"review_version":1}