{"id":"aee3a5ce-525a-466a-9a64-bffff2b1528d","arxiv_id":"2506.24029","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The regular representation of every Neretin group is factorial, yielding the first non-discrete simple group whose von Neumann algebra is a factor.","lead":"This paper proves that Neretin groups, which are simple non-discrete groups acting on the boundary of a tree, have a factorial regular representation: their group von Neumann algebra is a factor. It gives the first known example of a non-discrete simple group with this property, while introducing a general criterion that also applies to other families of groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof of Theorem A is internally consistent, and the secondary Proposition 1.13 gap does not affect the central claim.","rationale":"The reader's weakest_assumption was property (⋆_H), specifically the two geometric facts about the tree boundary and the rigid stabilizer index. On inspection, both are correctly handled in the manuscript: the disjoint-ball lemma follows from ultrametricity and the fact that a nontrivial local similarity has a point x with g(x) ≠ x; the index estimates are explicit, finite, and have the right growth. Lemma 1.7 and Theorem 1.9 also check out; the H ≤ ker Δ_G hypothesis is stronger than necessary but does not create a gap. The genuine defect found by the reader, Proposition 1.13, is confined to the Hecke-algebra applications of Section 1.4 and is not used in the proofs of Theorems A, B, C, or D. Hence the central claim about factoriality of the regular representation of Neretin groups is not affected. I therefore maintain the reader's CONDITIONAL verdict only because of the secondary Proposition 1.13 issue, not because of any obstacle to the main theorem.","tokens_in":24094,"tokens_out":43798,"duration_ms":465493,"concrete_test":"Verify the core estimate computationally for a concrete Neretin group, e.g., d = k = 2. For a fixed nontrivial local similarity g, such as the transposition of the two level-1 subtrees, enumerate the orbit of gK^{(n)} under O_{d,k}^{(n)} for n = 2,3,4,5 and recompute the lower bound [O^{(n-n0)}_{d,d}:K_{d,d}]. If the actual orbit size grows at least as fast as the stated index and the product c μ^2 diverges, the key step of Theorem 1.19 is corroborated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the central argument in Theorem 1.19, I find no load-bearing flaw. The claim requires property (⋆_H) for H = O_{d,k} with K_n = K_{d,k}^{(n)}. The proof supplies the lower bound c_{O_{d,k}}(gK^{(n)}) ≥ [H(w,n):L_w] where H(w,n) = rist_{O^{(n)}}(B_w) and L_w = rist_{K_{d,k}}(B_w), together with the index identity [H(w,n):L_w] = [O^{(n-n0)}_{d,d}:K_{d,d}], the bound [K_{d,d}:K^{(n-n0)}_{d,d}] ≤ (d!)^{d^{n-n0}}, and [K_{d,k}:K^{(n)}_{d,k}] ≤ k!(d!)^{k d^n}. These combine to make c_{O}(gK^{(n)}) μ(K^{(n)})^2 grow at least like (d^m/e)^{d^m}/(d!)^{d^m} times a constant, which diverges as n grows. The geometric claim that a nontrivial local similarity admits a ball B_w with g(B_w) ∩ B_w = ∅ is standard for ultrametric balls: if x ≠ g(x), choose a sufficiently small clopen ball around x avoiding g(x); contractions/expansions with a fixed point can be avoided by choosing the ball away from that fixed point. The modular-factor computation in Lemma 1.7 is also consistent, with the H ≤ ker Δ_G hypothesis not actually needed for the formal chain but harmless. The reader's identified issue in Proposition 1.13 concerns the Hecke-algebra applications in Section 1.4 and does not enter the proof of Theorem A, Theorem B, or Corollary C. Therefore I do not find a concern that would undermine the main factoriality result.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a factoriality criterion for the left regular representation of totally disconnected locally compact groups. Theorem B (Theorem 1.9) states that if a closed subgroup H of G lies in ker Δ_G and there is a compact-open neighborhood basis (K_n) such that limsup_n c_H(gK_n) μ_G(K_n)^2 = ∞ for every nontrivial g, then L(H)'∩L(G)=C. The main application is to Neretin groups N_{d,k}: with H=O_{d,k} and K_n=K_{d,k}^{(n)}, explicit rigid-stabilizer estimates are used to verify the criterion, yielding L(O_{d,k})'∩L(N_{d,k})=C. Theorem A (Theorem 1.19) then identifies L(N_{d,k}) and L(O_{d,k}) as type II_∞ factors, giving the first example of a non-discrete simple group whose regular representation is factorial. The paper also derives type consequences, applies the criterion to Hecke von Neumann algebras of amalgamated free products, to HNN extensions of locally compact groups, and proves a crossed-product analogue (Theorem D) with an application to the action of N_{d,k} on the visual boundary.","tokens_in":24391,"tokens_out":16565,"duration_ms":166963,"significance":"The main result is substantial and original. It settles the factoriality question for Neretin groups, identifies the type of the resulting factors, and provides the first known example of a non-discrete simple group with factorial group von Neumann algebra. The proof is elementary and self-contained, with explicit lower bounds and no fitted parameters or circularity; it relies only on standard external results. The general criterion Theorem B is likely to be useful beyond Neretin groups, and the crossed-product theorem gives new factorial type III examples. The secondary applications, especially the Hecke-algebra results for Burger-Mozes groups, are interesting but contain a proof gap discussed below.","major_comments":[{"comment":"The proof of Proposition 1.13 contains an unjustified step. Let N_A(K)_0 denote the kernel of the modular function restricted to N_A(K). The proof asserts that if |N_A(K)_0/K|≤2, then N_A(K)_0 is a compact open normal subgroup of N_A(K), \"therefore\" N_A(K)=N_A(K)_0. This implication is false: a compact open normal subgroup need not be the whole group. For a concrete example, take K=Z_p and A=K⋊Z with the Z-action by multiplication by p; then K is compact open in A, N_A(K)=A, N_A(K)_0=K, and |N_A(K)/K|=∞ while |N_A(K)_0/K|=1. The hypotheses of the proposition can be met in this situation by taking B=K×C_2, so |N_B(K)/K|=2. The subsequent construction of elements a_1,a_2∈N_A(K)_0\\K then fails in general. Since Proposition 1.8 requires the relevant subgroup to be contained in ker Δ_G, the reduction to the unimodular part is not justified without an additional hypothesis (for example, unimodularity of A and B, which does hold in the intended Burger-Mozes application where the vertex groups are compact). As stated, the proof of Proposition 1.13 does not establish the factoriality of p_K L(G)p_K, and Corollary 1.14 and Example 1.15 inherit this gap.","section":"§1.4, Proposition 1.13"}],"minor_comments":[{"comment":"Page 6 contains a duplicated phrase: \"endowed with endowed its Plancherel weight\" should be \"endowed with its Plancherel weight\".","section":"§1.1"},{"comment":"The notation \"N_A(K)_0 = N_A∩ker Δ_G(K)\" is ill-formed; the modular function is on G, so the intended meaning is presumably the kernel of Δ_G restricted to N_A(K), or equivalently the unimodular part of N_A(K). The definition should be stated unambiguously.","section":"§1.4, Proposition 1.13"},{"comment":"In the proof of the second part of Proposition 2.4, the expression \"[x−a(gK)=0\" is garbled; it should read \"\\widehat{x-a}(gK)=0\".","section":"§2.2, Proposition 2.4"},{"comment":"The geometric assertion that, for every nontrivial local similarity g, one can find a ball B_w with g(B_w)∩B_w=∅ is stated without proof; a one-sentence justification would improve readability, although the claim is standard and correct.","section":"§1.6, Theorem 1.19"}],"recommendation":"major_revision","confidential_remarks":"The main theorem and the central criterion appear sound: the estimates in Theorem 1.19 are explicit and check out, and the flagged gap in Proposition 1.13 does not enter the proof of Theorem A, Theorem B, or Theorem D. The issue in Proposition 1.13 is nonetheless a genuine mathematical error in a stated theorem and its applications, so I recommend major revision rather than rejection. The fix is likely local: adding a unimodularity hypothesis on A and B (or restricting to the compact-vertex case needed for Corollary 1.14) should repair the argument, since the Burger-Mozes application uses compact vertex stabilizers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper proves that Neretin groups have factorial left regular representation, the first example for a non-discrete simple group, and it does so with a genuinely new and elementary criterion. The main theorem and its supporting estimates are sound as far as I can tell. There is one real flaw, but it sits in a secondary application and does not touch the central result.\n\nWhat is new: Theorem B is a Fourier-coefficient criterion for factoriality of L(G) for tdlc groups, based on conjugation-orbit growth c_H(gK_n) μ(K_n)^2. The proof via averaging projections and the inequality in Lemma 1.7 is clean. The Neretin application is not a formality: it requires the explicit lower bound c_{O_{d,k}}(g K^{(n)}) μ(K^{(n)})^2 ≥ [(d^m/e)^{d^m}/(d!)^{d^m}] / [k!(d!)^{2kd^n}] → ∞. That estimate uses the rigid stabilizer index and the standard fact that a nontrivial local similarity can be pushed off a ball. I checked the index bookkeeping and the modular factor in Lemma 1.7; it is consistent. Corollary C's type argument is a nice byproduct, and Theorem D's crossed-product version is a natural extension. The paper is also honest about what it does not claim: Section 1.5 says the HNN consequences are essentially reformulations of Raum and Suzuki.\n\nThe soft spot is Proposition 1.13. The proof asserts that if N = N_A(K)_0 has quotient N/K of order ≤ 2, then N is a compact open normal subgroup of N_A(K), hence N_A(K) = N. That implication is false: a compact open normal subgroup can have infinite quotient, and the modular character can be infinite cyclic. The Hecke-algebra applications to amalgamated free products and Burger–Mozes groups rest on this proposition, so those corollaries are not justified as written. This looks repairable—one would need a different argument or an additional hypothesis—but it is a genuine gap, not a typo. It does not enter the proofs of Theorems A, B, or D.\n\nWho should read it: anyone working on type I/II questions for tdlc groups, or on Neretin groups and simple locally compact groups. The main theorem will be cited. I would send it to a serious referee: the central claim is important and the proof is detailed enough to verify. I would ask the referee to focus on fixing or isolating the Proposition 1.13 issue before publication.\n\nRecommended action: referee, with revision expected.","headline":"This paper proves factoriality of the regular representation of Neretin groups—the first such example for a non-discrete simple group—with a new and mostly sound criterion, though a real gap in a secondary Hecke-algebra application needs fixing.","tokens_in":24986,"tokens_out":3055,"would_cite":true,"duration_ms":35170,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L10","22D25","22D10","20E08"],"pacs":[],"model":"deepseek-v4-flash","headline":"The left regular representation of every Neretin group is factorial, giving the first non-discrete simple group whose von Neumann algebra is a factor.","keywords":["Neretin groups","factorial regular representation","von Neumann algebras","totally disconnected locally compact groups","tree almost automorphism groups","type II infinity factors","crossed products","Fourier coefficients on group von Neumann algebras"],"falsifier":"The claim is settled by calculating, for a nontrivial local similarity $g$ and the standard compact open subgroups $K^{(n)}$, whether $c_{O_{d,k}}(gK^{(n)}) \\mu(K^{(n)})^2$ really is unbounded; a single nontrivial $g$ for which this quantity stays bounded, or a pair $(d,k)$ for which the rigid-stabilizer index estimates fail, would break property $(\\star_{O_{d,k}})$ and with it the proof of Theorem A.","tokens_in":23808,"feed_emoji":"","tokens_out":8003,"duration_ms":82377,"temperature":0.7,"pith_summary":"This paper proves that the left regular representation of every Neretin group $N_{d,k}$ is factorial: the von Neumann algebra it generates has trivial center. Precisely, for the open amenable subgroup $O_{d,k}$ of local isometries, the commutant intersection $L(O_{d,k})' \\cap L(N_{d,k})$ is just the scalars. From this the author concludes that $L(O_{d,k})$ is the hyperfinite $\\mathrm{II}_{\\infty}$ factor and $L(N_{d,k})$ is a type $\\mathrm{II}_{\\infty}$ factor. Since Neretin groups are abstractly simple, this provides the first example of a non-discrete simple group with factorial regular representation. The proof is built on a new, elementary factoriality criterion for totally disconnected locally compact groups, expressed through conjugation-orbit sizes of compact open subgroups.","feed_headline":"Neretin groups yield first non-discrete simple group factor","feed_subtitle":"The proof, built on orbit-counting estimates for boundary balls, identifies L(N_{d,k}) as a type II∞ factor.","key_machinery":"The argument is carried by Fourier coefficients on totally disconnected group von Neumann algebras: for $x \\in L(G)$, the numbers $\\hat{x}(gK) = \\langle x\\xi_K, \\xi_{gK}\\rangle$, where $K$ is a compact open subgroup and $\\xi_A$ is the normalized indicator of a coset. These coefficients determine $x$ completely, and for $z \\in L(H)' \\cap L(G)$ they satisfy the key inequality $\\|z\\|_\\infty^2 \\geq c_H(gK)|\\hat{z}(gK)|^2$, where $c_H(gK)$ is the cardinality of the orbit of $gK$ under conjugation by the normalizer $N_H(K)$. If $H$ lies in the modular kernel and the orbit sizes grow fast enough relative to Haar measure, this inequality forces every Fourier coefficient of $z$ away from the identity to vanish, so $z$ is scalar. For Neretin groups the verification uses two tree-boundary facts: any nontrivial local similarity can be isotoped so that $g(B_w) \\cap B_w = \\varnothing$ for some boundary ball $B_w$, and the rigid stabilizer of that ball in $O_{d,k}$ is large enough, giving the estimate $c_{O_{d,k}}(gK^{(n)}) \\mu(K^{(n)})^2 \\geq [(d^{n-n_0}/e)^{d^{n-n_0}}/(d!)^{d^{n-n_0}}] / [k!(d!)^{2kd^n}] \\to \\infty$.","core_discovery":"The central discovery is that the inclusion of $L(O_{d,k})$ in $L(N_{d,k})$ is irreducible in a strong sense: every operator in $L(N_{d,k})$ commuting with all of $L(O_{d,k})$ is a scalar, so $L(O_{d,k})' \\cap L(N_{d,k}) = \\mathbb{C}$. Consequently $L(N_{d,k})$ is a factor, and since both groups are unimodular the factor is of type $\\mathrm{II}_{\\infty}$; because $O_{d,k}$ is amenable, $L(O_{d,k})$ is the hyperfinite $\\mathrm{II}_{\\infty}$ factor. The general engine is a criterion applying to any totally disconnected locally compact group $G$ with a closed subgroup $H \\leq \\ker \\Delta_G$: if there is a basis $(K_n)$ of compact open subgroups for which every nontrivial $g$ satisfies $\\limsup_n c_H(gK_n) \\mu_G(K_n)^2 = \\infty$, then $L(H)' \\cap L(G) = \\mathbb{C}$. The author verifies this condition for the Neretin group with $H = O_{d,k}$ through explicit growth estimates comparing rigid stabilizers of boundary balls with the indices of the compact open subgroups.","pith_inferences":["The paper leaves open whether $L(N_{d,k})$ is amenable; if a later argument established non-amenability, the result would give a simple totally disconnected group whose group factor is non-amenable, which would contrast with the amenable hyperfinite factor associated to $O_{d,k}$.","The criterion suggests a general route to factoriality for non-discrete groups: instead of approximating by discrete C*-simple groups, one can check orbit-growth estimates directly, making the method a candidate for other piecewise groups acting on Cantor sets.","Factoriality of the regular representation is strictly weaker than C*-simplicity, so the theorem should be read as evidence for, but not a proof of, C*-simplicity of Neretin groups, which remains open.","A testable extension would be to apply the same orbit-counting estimate to topological full groups of minimal Cantor actions, where the rigid stabilizer growth may be computable and could yield new factorial examples."],"forward_implications":["The group von Neumann algebra $L(N_{d,k})$ is a type $\\mathrm{II}_{\\infty}$ factor, and $L(O_{d,k})$ is the hyperfinite $\\mathrm{II}_{\\infty}$ factor, giving a new proof that both groups are not of type I.","The factoriality criterion applies to any closed subgroup of $N_{d,k}$ containing $O_{d,k}$, and to natural generalizations such as almost automorphism groups of regular branch groups, coloured Neretin groups, and piecewise full groups of profinite branch group actions.","For the visual boundary action, the crossed product $L^\\infty(\\partial T_{d,k}, \\nu) \\rtimes N_{d,k}$ is shown to be a type $\\mathrm{III}_{1/d}$ factor.","The same orbit-counting methods yield factoriality for certain HNN extensions, including profinite completions of Baumslag–Solitar groups, and identify some Hecke von Neumann algebras of Burger–Mozes groups as interpolated free group factors.","Since $N_{d,k}$ and $O_{d,k}$ are not type I and are non-discrete, the factors $L(N_{d,k})$ and $L(O_{d,k})$ cannot be of type I, so their type is determined as $\\mathrm{II}_{\\infty}$."],"supporting_citations":[{"why":"Introduces the Neretin groups, the objects whose regular representation is shown to be factorial.","marker":"[Ner92]"},{"why":"Supplies the unique locally compact topology on $N_{d,k}$ making the compact open subgroup $K_{d,k}$ open, which is needed to define the group von Neumann algebra.","marker":"[LB17]"},{"why":"Provides the standard facts on group von Neumann algebras, in particular $L(G) = \\rho_G(G)'$ and the Plancherel weight used throughout.","marker":"[Tak03]"},{"why":"Gives the classification of type III factors and the $S$-invariant used to determine the type of the crossed products.","marker":"[Con73]"},{"why":"Used to rule out type $\\mathrm{III}_0$ for the boundary crossed product, forcing it to be type $\\mathrm{III}_{1/d}$.","marker":"[CT77]"},{"why":"Establishes abstract simplicity of Neretin groups, which makes the factoriality result the first for a non-discrete simple group.","marker":"[Kap99]"},{"why":"Provides the earlier C*-simplicity criterion for totally disconnected groups via approximation by discrete C*-simple groups, the discrete analogue that the present W*-criterion extends.","marker":"[Suz17]"},{"why":"Shows C*-simplicity of profinite completions of Baumslag–Solitar groups, whose von Neumann algebra factoriality is recovered by the present methods.","marker":"[Rau21]"}],"fun_headline_variants":["First non-discrete simple group with factorial regular representation","Neretin groups yield factorial regular representation, type II∞","Type II∞ factor from Neretin groups' regular representation","Factoriality of Neretin groups: first non-discrete simple case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof stands or falls on the growth estimate $c_{O_{d,k}}(gK^{(n)}) \\mu(K^{(n)})^2 \\to \\infty$ for every nontrivial $g$; this requires both that every nontrivial local similarity can be isotoped so that $g(B_w) \\cap B_w = \\varnothing$ for some boundary ball $B_w$, and that the rigid stabilizer of a boundary ball grows at the asserted factorial rate, so if either geometric fact fails for even one nontrivial element the lim sup condition in $(\\star_H)$ fails and the criterion no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["First non-discrete simple group with factorial regular representation","Neretin groups yield factorial regular representation, type II∞","Type II∞ factor from Neretin groups' regular representation","Factoriality of Neretin groups: first non-discrete simple case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000882,"raw_usage":{"total_tokens":3769,"prompt_tokens":862,"completion_tokens":2907,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":2837}},"tokens_in":478,"tokens_out":2907,"duration_ms":19930,"temperature":1.0,"reasoning_tokens":2837,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:29:18.264177+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim is settled by calculating, for a nontrivial local similarity $g$ and the standard compact open subgroups $K^{(n)}$, whether $c_{O_{d,k}}(gK^{(n)}) \\mu(K^{(n)})^2$ really is unbounded; a single nontrivial $g$ for which this quantity stays bounded, or a pair $(d,k)$ for which the rigid-stabilizer index estimates fail, would break property $(\\star_{O_{d,k}})$ and with it the proof of Theorem A.","supporting_citations":[],"review_version":1}