{"id":"1b1349bc-65ee-4532-b69b-a043c8cd3afc","arxiv_id":"2510.23864","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For split actions of countable groups on abelian Lie-type groups, faithful dually ergodic or doubly ergodic linear part plus infinite orbits or finite stabilizers on the discrete part makes L(N⋊Γ) a factor; and a counterexample shows ergodicity of a linear action does not imply ergodicity of its inv","lead":"This paper gives conditions under which the group von Neumann algebra of a semidirect product N⋊Γ is a factor, where N is an abelian group of Lie type and Γ is countable discrete, and it supplies new examples. It also constructs, for every n≥3, a countable subgroup of SL(n,R) acting ergodically on R^n whose inverse-transpose (dual) action is not ergodic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's proof invokes [Sch95, Cor. 5.4] to pass from essential range R to ergodicity of a skew product over an infinite-measure base with noncompact target, without verifying the corollary's hypotheses; if that result requires a probability-preserving base or compact target, the central countere","rationale":"The reader's weakest assumption identifies exactly the most load-bearing concern: the proof of Theorem A depends on a cited theorem whose hypotheses are not verified for the infinite-measure, noncompact-target setting. This is a genuine gap in the written argument and could invalidate the paper's central new counterexample if the cited corollary is not applicable. However, the theorem statement itself appears true — a direct ergodicity proof for Γ0↷R^n is straightforward, and the rest of the paper's factoriality results do not rely on this particular step. Thus the paper's main claims are likely correct, but the exposition needs a verification of the essential-range theorem's hypotheses or a direct proof. The reader's verdict CONDITIONAL is appropriate; no verdict change is needed. Other flagged issues (undefined theorem reference, dimension typo in Proposition 5.11, unjustified assertion in Corollary 5.6) are secondary and do not affect the central argument as seriously.","tokens_in":18856,"tokens_out":26480,"duration_ms":231971,"concrete_test":"Check the exact statement of [Sch95, Corollary 5.4] (or [Sch77, Corollary 5.4]) to see whether it applies to a non-singular ergodic action on a σ-finite infinite-measure space with a Borel cocycle taking values in noncompact abelian R. If the corollary requires an invariant probability measure or compact target, then Proposition 4.5 is missing a key hypothesis and Theorem A is not proven as written. Alternatively, give a direct proof of ergodicity of Γ0↷R^n for n=3 without invoking [Sch95]: for a.e. v∈R^2 the set {w·v : w∈Z^2} is dense in R, so any Γ0-invariant f∈L∞ is independent of the first coordinate for a.e. v; SL(2,Z)-ergodicity on R^2 then forces f constant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Proposition 4.5 the paper proves ergodicity of Γ0↷R^n by decomposing R^n = R × R^{n-1}. The action is the skew product of α: Γ0↷R^{n-1} (where Γ0 contains SL(n-1,Z)) with cocycle c(g,v)=w^T v. The paper shows α is ergodic and E(c)=R, then cites [Sch95, Corollary 5.4] to conclude the skew action is ergodic. That is the decisive step for Theorem A. However, the paper nowhere states the hypotheses of [Sch95, Cor. 5.4] (or the related [Sch77, Cor. 5.4] used in §2.2). In the standard version of the essential-range theorem the base is an ergodic measure-preserving action on a probability space and the target is a locally compact second countable group (often compact). Here the base is R^{n-1} with infinite Lebesgue measure and the target is R (noncompact). If the corollary indeed requires a finite invariant measure on the base, the proof does not go through. This is not merely a cosmetic omission: Theorem A is the paper's central new counterexample, and no independent proof of ergodicity of Γ0↷R^n is given. A direct argument is available — for a.e. v the translations {w·v} are dense in R, so an invariant function cannot depend on the first coordinate, and then SL(n-1,Z)-ergodicity forces constancy — but the paper does not provide it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies factoriality of the group von Neumann algebra L(N⋊Γ) when N is an abelian locally compact group of Lie type and Γ is countable discrete. By Pontryagin duality, L(N⋊Γ) is identified with the crossed product L^∞(\\hat N)⋊Γ, so factoriality is analyzed through ergodicity and essential freeness of the dual action. The main results are: (Theorem A, proved in Proposition 4.5) for every n≥3, the group Γ0={A∈SL(n,Z): Ae1=e1} acts ergodically on R^n while its transpose Γ0^T acts non-ergodically, giving a negative answer to the paper's Question 1.1; and (Theorem B, realized in Theorems 5.2, 5.5, 5.9, 5.10) sufficient conditions for factoriality when N=R^m×D and the action splits, with corollaries covering actions of lattices and dense subgroups of SL(n,R), torsion discrete quotients, and Bernoulli-type discrete parts. The paper also contains criteria for actions on discrete abelian groups and examples involving SL(2,Z) subgroups.","tokens_in":19275,"tokens_out":15749,"duration_ms":156102,"significance":"If the main claims hold, Theorem A is a striking counterexample: ergodicity of a linear action on R^n is not invariant under the transpose automorphism for n≥3, despite being true for n=2. Theorem B provides a broad, explicit class of factorial group von Neumann algebras, including II_∞ examples and, via remark 5.12, non-injective factors from SL(n,R)-lattices. The paper is a pure theorem paper: there are no fitted parameters or numerical experiments, and the statements are constructive and checkable from standard results in ergodic theory and group duality. The overall strategy—using the essential range of a cocycle and applying Glasner–Weiss or Schmidt–Walters—is sound in principle. However, the current manuscript contains a load-bearing gap in the proof of Theorem A (the unstated hypotheses of the Schmidt essential-range theorem) and an undefined theorem reference in the proof of the necessity half of Theorem 5.2. These issues are local and fixable, but they must be addressed before the claims can be accepted.","major_comments":[{"comment":"Proposition 4.5's proof of ergodicity of Γ0↷R^n is the decisive step for Theorem A. The proof computes that the base action α on R^{n-1} is ergodic and that the essential range of the cocycle c is R, then invokes [Sch95, Corollary 5.4] (and in §2.2 [Sch77, Corollary 5.4]) to conclude that the skew product is ergodic. However, the paper nowhere states the hypotheses of those corollaries. Here the base is R^{n-1} with infinite Lebesgue measure and the target is the noncompact group R. If Schmidt's result is stated only for probability-preserving bases or compact targets, the argument does not go through as written. Please state the theorem in full and verify its hypotheses, or supply a direct proof (for instance, using density of {w·v} for v in a co-null set and SL(n−1,Z)-ergodicity on R^{n-1}).","section":"§4.5 / §2.2"},{"comment":"The converse part of Theorem 5.2, which supports the necessity claims in Theorem B, is justified by an undefined reference: 'by Theorem Thm: double ergodicity and cocycle superrigid implies full action ergodic'. No such theorem appears in the paper or in the reference list. The proof is incomplete as written. This should be replaced by a concrete argument or citation—for example, that factoriality of L∞(R^m×\\hat D)⋊Γ implies ergodicity of the underlying dual action, together with Proposition 3.3 for the orbit condition.","section":"§5.2, proof of Theorem 5.2"}],"minor_comments":[{"comment":"The sentence 'Γ0 acts trivially on the first copy of R' is false: the action sends (x,v) to (x + w^T v, Av), so the first coordinate is moved by the cocycle. What is meant is that R^n is realized as a skew product over the base R^{n-1}. Please rephrase.","section":"§4.5, Proposition 4.5"},{"comment":"The notation 'g·F_w = F_w' is ambiguous: under the skew action the first coordinate changes, so the equality can only refer to the base action α_g. Clarify that α_g fixes F_w pointwise for the chosen g.","section":"§4.5, Proposition 4.5"},{"comment":"The statement writes L((R^n × ⊕_Γ Λ)⋊Γ), but Γ is a finite-index subgroup of SL(2,Z), so the linear action is on R^2. Either replace R^n by R^2 or adjust the hypothesis to a lattice in SL(n,R) for the appropriate n.","section":"§5, Proposition 5.11"},{"comment":"There are small wording/typos: Theorem 5.5 condition 2 says 'doubly dually ergodic' instead of 'dually doubly ergodic', and Corollary 5.10 says 'Φ|Γ is faithful' where it should say 'Φ|_{N^∘}' or 'η is faithful'. Please correct.","section":"§5.2, Theorem 5.5 and Corollary 5.10"},{"comment":"In the statement of Lemma 4.1, 'SL(n,R^n)' should be 'SL(n,R)'.","section":"§4.1, Lemma 4.1"},{"comment":"The notation E(ω) is used for both the essential-value set in the one-point compactification and for its intersection with T; consider distinguishing these to avoid confusion.","section":"§2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is promising and the main ideas are likely correct, but the current version needs two essential repairs: (1) a full statement and verification of the Schmidt essential-range theorem used in Proposition 4.5, or a self-contained ergodicity proof for Γ0↷R^n; and (2) a proper justification for the converse part of Theorem 5.2, replacing the undefined 'Theorem Thm' label. I would also check that no other dangling cross-references remain. The examples and corollaries appear valuable, and the issues are fixable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is genuine. For each n≥3, the subgroup Γ0 = {A∈SL(n,Z) : Ae1=e1} gives a linear action on R^n that is ergodic while its transpose action is not. The construction is simple and the essential-range computation is convincing. I think the proof goes through; the direct argument is available anyway since for a.e. v the translations {w·v} are dense in R, so an invariant function cannot depend on the first coordinate, and then SL(n-1,Z)-ergodicity of the base forces constancy. The paper's reliance on [Sch95, Cor. 5.4] without stating its hypotheses is a real gap in presentation—the stress-test concern about infinite-measure base and noncompact target is worth taking seriously, but it does not invalidate the result.\n\nThe factoriality criteria in Section 5 are useful. The split-action theorem (Theorem 5.2) is a clean combination of Glasner-Weiss weak mixing and the Schmidt-Walters mixing criterion, and the torsion-group variant (Theorem 5.5) gives a large class of examples without assuming the action splits. The authors are honest about the split-action restriction and correctly point to recent work by Morando and Miyamoto for nearby territory. The citation pattern is fine: the reference to [BCDK24] is to a published independent result, not to the present paper.\n\nThe soft spots are real but mostly cosmetic. Theorem 5.2's converse cites a nonexistent 'Theorem Thm: double ergodicity and cocycle superrigid implies full action ergodic'; that should be replaced by the obvious fact that for a diagonal product action, ergodicity of the product implies ergodicity of each factor. Proposition 5.11 has a dimension error—a subgroup of SL(2,Z) acts on R^2, not R^n, so the statement should be R^2. Corollary 5.6's proof is a one-liner and contains an R^m/R^n typo; the result is a direct consequence of Theorem 5.5, so the fix is straightforward. There are also a handful of small typos ('reamins', 'Z^d') that a careful revision should catch.\n\nNone of these issues undermines the central claims. The paper deserves a serious referee and should be accepted after a cleanup. I would bring it to a reading group if anyone cares about factorial locally compact group von Neumann algebras or ergodic linear actions.","headline":"Theorem A is a real counterexample and the paper is worth refereeing, but the write-up needs a cleanup pass: a broken reference, a dimension typo, and an underspecified citation in the key proof.","tokens_in":19766,"tokens_out":4236,"would_cite":true,"duration_ms":46008,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A15","46L10","22D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For n≥3, ergodicity of a linear action on R^n is not preserved under taking transposes of the acting group.","keywords":["ergodic actions","group von Neumann algebras","factoriality","linear actions on R^n","dual action","semidirect products","locally compact abelian groups","essential range of cocycles"],"falsifier":"Check whether the corollary of Schmidt's theorem quoted for the skew product requires the base action to be probability-preserving and the cocycle target to be compact. If it does, the essential-range computation E(c)=R alone is insufficient, and a direct search for a Γ0-invariant non-null-non-conull subset of R^n would settle ergodicity. Additionally, testing the Γ0-action on bounded measurable functions that depend on the first coordinate could reveal an invariant function.","tokens_in":18760,"feed_emoji":"","tokens_out":5619,"duration_ms":53702,"temperature":0.7,"pith_summary":"The paper establishes that ergodicity of a measure-preserving linear action of a countable group on R^n does not imply ergodicity of the dual action (obtained by transposing and inverting the matrices), for every n≥3. It constructs a concrete subgroup Γ0 of SL(n,Z) whose linear action is ergodic but whose dual action is not. The paper then gives sufficient conditions – split action, faithful dually doubly ergodic connected part, and infinite orbits on the discrete quotient – under which the group von Neumann algebra L(N⋊Γ) is a factor for abelian Lie-type groups N. These conditions yield large families of examples, including actions of lattices and dense subgroups with torsion or orbit-infinite discrete parts.","feed_headline":"A subgroup of SL(n,Z) acts ergodically; its transpose doesn't","feed_subtitle":"The counterexample yields new factorial group von Neumann algebras from semidirect products.","key_machinery":"The main objects are the subgroup Γ0={A∈SL(n,Z):Ae1=e1}, the skew-product decomposition of its linear action over the ergodic action on R^{n-1} with a Z^{n-1}-valued cocycle, and the essential-range criterion for skew-product ergodicity. For the factoriality theorems, the key machinery is the isomorphism L(N⋊Γ)≅L^∞(N̂)⋊Γ, the diagonal (split) action decomposition N=N^0×N/N^0, and the use of double ergodicity (weak mixing) or mixing of one component to force ergodicity of the diagonal dual action.","core_discovery":"The central claim is an asymmetry: for n≥3, the map A↦(A^T)^{-1} on SL(n,R) is not inner, and the subgroup Γ0={A∈SL(n,Z):Ae1=e1} realizes the asymmetry. Its linear action on R^n is ergodic, proved by writing it as a skew product over the ergodic action of Γ0 on R^{n-1} and showing the associated cocycle has essential range all of R. Its dual action is non-ergodic because Γ0^T fixes the character e1, giving a non-scalar invariant function. The paper further proves that factoriality of L(N⋊Γ) for abelian Lie-type N without compact connected subgroups follows from a split action whose connected-component action is faithful and dually doubly ergodic and whose discrete-component action has infini","pith_inferences":["The non-equivalence of ergodicity under transpose suggests that no purely spectral or rigidity-based criterion for ergodicity of linear actions can be invariant under this automorphism, so the search for an intrinsic characterization may need orbit-equivalence or cocycle invariants.","The essential-range computation may generalize to other fixed-character subgroups, potentially giving more counterexamples in SL(n,R).","The split-action condition is probably removable in greater generality: the paper's own torsion case (Theorem 5.5) shows the twisted cocycle vanishes under a purely algebraic hypothesis; one might test whether finite-generation or cohomological vanishing of the discrete part achieves the same.","The type II_∞ examples here suggest a systematic route to factorial locally compact groups with prescribed modular behavior, using non-measure-preserving actions as the paper hints."],"forward_implications":["Question 1.1 (ergodicity of an action iff ergodicity of its dual) has a negative answer in every dimension ≥3.","Any intrinsic characterization of ergodic countable subgroups of SL(n,R) cannot be invariant under the automorphism A↦(A^T)^{-1}.","For n≥3, lattices and countable dense subgroups Γ