{"id":"c0866a15-4569-417f-bb32-bc48d802a828","arxiv_id":"2506.08755","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":1.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.","lead":"This survey reviews what can be learned about an infinite group from the collection of its finite quotient groups. It collects examples of non-isomorphic groups with identical profinite completions and identifies the small number of properties that are determined by the completion.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the survey's conditional use of Serre's CSP is explicit and does not undercut its central claim.","rationale":"The reader's verdict of ACCEPT is sound. The identified weakest assumption, Serre's conjecture on CSP, is real but applies mainly to the conditional classification theorems rather than to the survey's central descriptive claim. The main unconditional examples, especially the spinor groups with Witt index at least two, already establish that several natural properties are not profinite. The survey is careful to mark conditional statements, noting exactly which exceptional cases depend on CSP. Consequently, even if Serre's conjecture were false in some higher-rank case, the survey's core narrative would survive; only the sharp classification statements would need revision. The proofreading errors are minor and do not change the mathematical content. No manufactured objection is warranted.","tokens_in":19991,"tokens_out":8426,"duration_ms":110559,"concrete_test":"Verify the p=2 case of Lemma 3.1 by symbolically computing whether the displayed 4x4 matrix B satisfies B^T diag(1,1,1,1) B = -diag(1,1,1,1) over Z_2; if the identity fails, the primary Spin(q7,2)/Spin(q3,6) example would need re-examination, but if it passes, the central engine of the negative results is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the survey's central claim that profinite properties are rare. The core negative examples do not rest on Serre's open conjecture: the paper states in Section 3 that spinor groups of non-degenerate integral quadratic forms with Witt index at least two have the congruence subgroup property, so Spin(q7,2)(Z) and Spin(q3,6)(Z) have isomorphic profinite completions unconditionally, and the contrasts in bounded cohomology and higher L2-Betti numbers depend on this pair and its S-arithmetic variants. Serre's conjecture only enters the classification theorems in Sections 3.8 and 3.9, where the paper explicitly labels which statements are conditional and which exceptions would be removed if the conjecture holds. A survey may legitimately report the frontier of a field, including results that are conditional on a widely believed open conjecture, as long as the conditionality is stated; here it is stated clearly. The proofreading slips, such as the garbled isomorphism chain in the proof of Lemma 5.1, appear typographical and do not affect the substance of the arguments surveyed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of recent work on the question which properties of finitely generated residually finite groups are determined by their profinite completion. It focuses on S-arithmetic groups and branch groups. The main thesis is that profinite properties are rare: the survey collects many constructions of non-isomorphic groups with isomorphic profinite completions (spinor groups, Chevalley groups, branch groups yielding uncountably many Grothendieck pairs) and identifies the few properties that are profinite (laws, polynomial growth, uniform amenability, the first ℓ2-Betti number among Lücky groups, and the sign of the Euler characteristic for S-arithmetic groups with the congruence subgroup property). The paper also states several open problems. The exposition includes proofs of standard results and clearly marks which quoted theorems are conditional on Serre's congruence subgroup conjecture.","tokens_in":20157,"tokens_out":7399,"duration_ms":74683,"significance":"The survey provides a valuable and current map of the field. Its main claims are supported by a coherent body of examples and by the explicit distinction between unconditional results and those depending on Serre's conjecture. The paper is particularly strong in highlighting the contrast between higher-rank S-arithmetic and branch-group constructions, and in listing the known profinite properties. A minor caveat is that a large fraction of the surveyed theorems are the authors' own results, which is natural in a survey of this area but means the selection is closely tied to their research program. Overall, if the manuscript is polished, it will be a useful reference for graduate students and researchers.","major_comments":[],"minor_comments":[{"comment":"In the proof of Lemma 5.1, the displayed chain 'bH2 ∼= H2 ∼= H1 ∼= bH2' is garbled; it should state that the profinite completions of H1 and H2 are isomorphic, for example 'bH2 ≅ Ψ(H1) ≅ bH1'.","section":"§5.1"},{"comment":"The notation 'cΓn' in the discussion of the groups Γn± is missing a hat; it should be \\widehat{\\Gamma_n^\\pm}.","section":"§3.2"},{"comment":"In Lemma 2.4, the displayed isomorphism 'bΓab ∼= [(Γab)' is typeset incorrectly; it should read \\widehat{\\Gamma^{ab}} \\cong (\\widehat{\\Gamma})^{ab}.","section":"§2.1"},{"comment":"The phrase 'not even a profinte invariant' contains a typo; it should be 'profinite'.","section":"§3.10"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a survey and draws heavily on the authors' own recent work, which is appropriate given their role in the area. The editor may wish to verify that the venue is appropriate for a survey. The local proof and typesetting errors listed in the minor comments should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"For a survey, this one does its job: it gives a clear, well-organized map of what is known about which group properties are profinite, with emphasis on S-arithmetic groups, branch groups, and their relatives. There are no new theorems, and the authors don't claim otherwise. What is genuinely useful is the synthesis: the unconditional negative examples (Spin(q7,2)(Z) vs Spin(q3,6)(Z) for bounded cohomology and higher ℓ²-Betti numbers; Aka's property (T) examples; the branch-group constructions with 2^ℵ₀ Grothendieck pairs), clearly separated from the positive results (laws, polynomial growth, uniform amenability, first ℓ²-Betti number among Lücky groups, sign of the Euler characteristic for S-arithmetic groups with CSP). The conditional material on Serre's congruence subgroup property is flagged honestly, and the core 'profinite properties are rare' claim does not depend on Serre's conjecture, so the spine of the survey holds up.\n\nThe soft spots are mostly cosmetic. The proof of Lemma 5.1 contains a garbled isomorphism chain ('bH2 ≅ H2 ≅ H1 ≅ bH2' presumably should be bH1 ≅ bH2), and there are stray hats missing in Section 3.2 and in the proof of Theorem 5.8. These are typos, not mathematical errors, but they should be fixed before publication. The self-citation load is heavy but legitimate: the authors are the main contributors to several of the subareas surveyed, and the cited results are real. One can complain that the selection is skewed toward their own work, but they say up front that the focus is on S-arithmetic groups, branch groups, and relatives, so that is a stated editorial choice rather than a hidden bias.\n\nWho gets value from this: a graduate student or a researcher in geometric group theory wanting a quick, reliable entry point into profinite rigidity; also someone looking for open problems (e.g., Problem 5.9 on first ℓ²-Betti number for all f.g. residually finite groups, Question 4.8 on Grothendieck pairs with property (T)).\n\nI'd send it to a serious referee. The typesetting slips are minor and easily fixed; the survey is honest, accurate, and fills a real gap in the literature. Recommendation: accept with minor revisions.","headline":"A reliable survey of profinite rigidity, indispensable for newcomers, with minor proofreading slips and a legitimate self-citation load.","tokens_in":20742,"tokens_out":4969,"would_cite":true,"duration_ms":46958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20E18","20E26","20F65","20G30","22E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This survey argues that for finitely generated residually finite groups, the profinite completion determines only a short list of properties, while almost every other structural feature is invisible to the finite quotients.","keywords":["profinite completion","profinite rigidity","S-arithmetic groups","branch groups","Grothendieck pairs","ℓ2-Betti numbers","congruence subgroup property","uniform amenability"],"falsifier":"Compute the profinite completion of $\\mathrm{SL}_3(\\mathbb{Z}[1/p])$ for a prime $p$ and compare it with the product of the local groups $\\mathrm{SL}_3(\\mathbb{Z}_\\ell)$; a finite quotient distinguishing the two would falsify the finite-congruence-kernel description behind the arithmetic rigidity results. A differently targeted check: find two finitely generated residually finite groups with isomorphic profinite completions where exactly one has polynomial word growth, which would refute the profiniteness of laws and polynomial growth.","tokens_in":19731,"feed_emoji":"","tokens_out":12645,"duration_ms":139841,"temperature":0.7,"pith_summary":"This survey examines how much of a finitely generated residually finite group is determined by its set of finite quotients, which is the same information as its profinite completion. The paper's central thesis is that profinite properties are rare: bounded cohomology, higher $\\ell^2$-Betti numbers, property (T), the fixed-point property FA, finiteness properties, the centre, and torsion-freeness all fail to be profinite, with explicit non-isomorphic groups sharing a profinite completion for each. The properties known to be profinite are short and specific: satisfying a law, polynomial word growth, uniform amenability, the first $\\ell^2$-Betti number among groups satisfying approximation in degree one, and, within arithmetic classes with finite congruence kernel, the sign of the Euler characteristic and the covolume. The survey therefore gives a map of what finite quotients can and cannot reveal about an infinite group, together with the open problems, including the profinite rigidity of non-abelian free groups and of $\\mathrm{SL}_n(\\mathbb{Z})$ for $n \\ge 3$.","feed_headline":"Few group properties survive the profinite completion","feed_subtitle":"A survey shows that laws and first ℓ2-Betti numbers are profinite, while amenability, property (T), and torsion are not.","key_machinery":"The profinite completion $\\widehat{\\Gamma} = \\varprojlim \\Gamma/N$ is the inverse limit of all finite quotients, and two finitely generated residually finite groups have the same finite quotients exactly when their completions are isomorphic. Negative examples come from two sources: the congruence subgroup property, which makes $\\widehat{\\Gamma}$ a product of local factors so that local isomorphisms between number fields or algebraic groups can be traded without changing the completion; and branch-group and telescope constructions, where the completion is an iterated (semi)direct product depending only on low-level data, giving Grothendieck pairs. Positive results use word maps, whose continuity forces laws to lift from $\\Gamma$ to $\\widehat{\\Gamma}$; the approximation formula expressing the first $\\ell^2$-Betti number as a limit of finite-index Betti numbers; and Euler-characteristic and covolume computations that are controlled by the congruence subgroup property. The load-bearing named objects are the congruence kernel, the adelic superrigidity principle, and the iterated wreath product.","core_discovery":"The central claim, on the paper's own terms, is that the profinite completion of a finitely generated residually finite group keeps only a thin algebraic and analytic skeleton of the group. The negative half is carried by two construction principles: for $S$-arithmetic groups with finite congruence kernel, the completion factorises over local places, so locally isomorphic quadratic forms or number fields can be exchanged to produce non-isomorphic groups with the same completion; and for branch groups built by spinal actions, the completion is an iterated wreath product depending only on first-level data, yielding $2^{\\aleph_0}$ Grothendieck pairs, i.e. inclusions $\\Delta \\subsetneq \\Gamma$ of a proper subgroup that induce an isomorphism of profinite completions. In particular, one member of such a pair can be amenable while the other contains a non-abelian free group, so amenability is not profinite. The positive half lists exactly what does survive: laws, polynomial growth, uniform amenability, the first $\\ell^2$-Betti number under approximation in degree one, and, among arithmetic groups with the congruence subgroup property, the sign of the Euler characteristic and equality of covolumes.","pith_inferences":["The survey leaves implicit that the congruence subgroup property is the single structural bottleneck: if it were proved in full, the exception lists in the solitary-lattice theorems would shrink, and the corresponding classifications would become unconditional rather than conditional.","A testable extension of the branch-group method is to use the telescope construction on other iterated semidirect products, potentially producing 2-generated groups with prescribed profinite completions and thereby turning many finitely generated profinite groups into profinite completions of finitely generated discrete groups.","The listed profinite properties are all semidecidable from an enumeration of the finite quotients, while the non-profinite properties are not; a natural sharpening of the survey's dichotomy is to ask which profinite properties are recursively computable from the finite quotient data, a question the paper does not address.","Given the survey's evidence that torsion and homology are invisible to the completion, the open problem of whether the first $\\ell^2$-Betti number is profinite for all finitely generated residually finite groups is likely to have a negative answer; the non-approximable torsion groups with positive first $\\ell^2$-Betti number are the natural place to look for a counterexample."],"forward_implications":["Laws transfer to the profinite completion and back, so any property definable by a law, including being abelian, nilpotent, solvable, or virtually such, is profinite.","Polynomial word growth is profinite, so a finitely generated residually finite group has polynomial growth exactly when its profinite completion satisfies the corresponding law-theoretic condition.","Uniform amenability is profinite even though amenability is not; the branch-group constructions yield $2^{\\aleph_0}$ Grothendieck pairs in which the proper subgroup is amenable and the ambient group contains a non-abelian free subgroup.","The first $\\ell^2$-Betti number is determined by the profinite completion for all groups satisfying approximation in degree one, hence in particular for all finitely presented groups.","Among $S$-arithmetic groups with finite congruence kernel, profinite commensurability preserves the sign of the Euler characteristic and, within a fixed Lie group, the covolume; consequently finite-volume hyperbolic 3-manifolds with profinitely isomorphic fundamental groups would have equal volume if the relevant arithmetic cases are representative."],"supporting_citations":[{"why":"Establishes that two finitely generated residually finite groups have the same finite quotients iff their profinite completions are isomorphic, which frames the whole guiding question.","marker":"[12]"},{"why":"Introduces the quadratic-form local-isometry trick showing that $\\mathrm{Spin}(q_{7,2})(\\mathbb{Z})$ and $\\mathrm{Spin}(q_{3,6})(\\mathbb{Z})$ are profinitely isomorphic but not isomorphic, the template for the negative $S$-arithmetic examples.","marker":"[1]"},{"why":"Proves adelic superrigidity, giving the equivalence between profinite commensurability and local isomorphism of algebraic groups that underlies the solitary classifications.","marker":"[26]"},{"why":"Constructs $2^{\\aleph_0}$ Grothendieck pairs with an amenable member and a non-amenable member, and proves uniform amenability is profinite.","marker":"[35]"},{"why":"Classifies profinitely solitary split simple arithmetic groups under the congruence subgroup property, identifying exactly which groups are determined by profinite commensurability.","marker":"[33]"},{"why":"Constructs families of non-isomorphic and non-commensurable lattices with isomorphic completions, showing that profinite rigidity fails broadly in higher-rank Lie groups.","marker":"[28]"},{"why":"Proves that the sign of the Euler characteristic is a profinite invariant for arithmetic groups with the congruence subgroup property.","marker":"[30]"},{"why":"Proves equality of covolumes for profinitely commensurable higher-rank lattices with finite congruence kernel, giving the volume profiniteness result.","marker":"[29]"}],"fun_headline_variants":["Profinite completion keeps only a thin skeleton of a group","Profinite view cannot see amenability","Many non-isomorphic groups share one profinite completion","Profinite completion cannot distinguish some groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the conjecture that every higher-rank $S$-arithmetic subgroup of a simple algebraic group has finite congruence kernel; if this fails for any such group, the profinite completions used in the solitary-group and lattice classifications no longer have the advertised product structure and those theorems would need revision.","fun_headline_variants_meta":{"raw":{"variants":["Profinite completion keeps only a thin skeleton of a group","Profinite view cannot see amenability","Many non-isomorphic groups share one profinite completion","Profinite completion cannot distinguish some groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000683,"raw_usage":{"total_tokens":3031,"prompt_tokens":807,"completion_tokens":2224,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":2166}},"tokens_in":423,"tokens_out":2224,"duration_ms":22344,"temperature":1.0,"reasoning_tokens":2166,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:03:26.299889+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the profinite completion of $\\mathrm{SL}_3(\\mathbb{Z}[1/p])$ for a prime $p$ and compare it with the product of the local groups $\\mathrm{SL}_3(\\mathbb{Z}_\\ell)$; a finite quotient distinguishing the two would falsify the finite-congruence-kernel description behind the arithmetic rigidity results. A differently targeted check: find two finitely generated residually finite groups with isomorphic profinite completions where exactly one has polynomial word growth, which would refute the profiniteness of laws and polynomial growth.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that two finitely generated residually finite groups have the same finite quotients iff their profinite completions are isomorphic, which frames the whole guiding question."},{"cited_title":"Aka,Profinite completions and Kazhdan’s property (T), Groups Geom","cited_arxiv_id":null,"evidence_quote":"Introduces the quadratic-form local-isometry trick showing that $\\mathrm{Spin}(q_{7,2})(\\mathbb{Z})$ and $\\mathrm{Spin}(q_{3,6})(\\mathbb{Z})$ are profinitely isomorphic but not isomorphic, the template for the negative $S$-arithmetic examples."},{"cited_title":"Kammeyer and S","cited_arxiv_id":null,"evidence_quote":"Proves adelic superrigidity, giving the equivalence between profinite commensurability and local isomorphism of algebraic groups that underlies the solitary classifications."},{"cited_title":"Kionke and E","cited_arxiv_id":null,"evidence_quote":"Constructs $2^{\\aleph_0}$ Grothendieck pairs with an amenable member and a non-amenable member, and proves uniform amenability is profinite."},{"cited_title":"Kammeyer and R","cited_arxiv_id":null,"evidence_quote":"Classifies profinitely solitary split simple arithmetic groups under the congruence subgroup property, identifying exactly which groups are determined by profinite commensurability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs families of non-isomorphic and non-commensurable lattices with isomorphic completions, showing that profinite rigidity fails broadly in higher-rank Lie groups."},{"cited_title":"Kammeyer, S","cited_arxiv_id":null,"evidence_quote":"Proves that the sign of the Euler characteristic is a profinite invariant for arithmetic groups with the congruence subgroup property."},{"cited_title":"Profiniteness of higher rank volume","cited_arxiv_id":"2412.13056","evidence_quote":"Proves equality of covolumes for profinitely commensurable higher-rank lattices with finite congruence kernel, giving the volume profiniteness result."}],"review_version":1}