{"id":"dc6b2790-c150-4750-8638-3f77409674a5","arxiv_id":"2507.21862","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every irreducible non-uniform lattice in a higher-rank semisimple group of characteristic not 2 is character rigid.","lead":"The paper proves character rigidity for all non-uniform higher-rank lattices in semisimple groups of characteristic other than 2, confirming a conjecture of Stuck and Zimmer in this case. This means every such group's conjugation-invariant functions are either built from finite-dimensional representations or vanish off the center, and it forces strong rigidity of group actions and invariant random subgroups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof is internally coherent; its exposure is the quoted Raghunathan–Venkataramana theorem, exactly as the Reader flagged.","rationale":"I traced the main argument through §5–§7. The proof of Theorem 6.1 is internally coherent: the decomposition of traces into induced and congruence parts in Proposition 5.6, the projection argument using Lemma 6.7, and the two reductions in §7 all line up. The potentially delicate point that the projections p_n,q_n lie in π_φ(Γ)'' is supplied by amenability of the congruence unipotent groups: Følner averages converge to the invariant projection inside the subgroup von Neumann algebra. Lemma 6.7 is valid as stated; the second application of the trace argument works with the roles of p and q exchanged. In the first proof of character rigidity, the use of negative powers of a is legitimate because Theorem 6.1 also applies to a^{-1}. The paper is honest in Remark 6.3 that the characteristic-two restriction is imposed solely by the Raghunathan–Venkataramana theorem, and it does not overclaim otherwise. The Reader's weakest assumption identifies the same load-bearing external premise, and the verdict ACCEPT remains appropriate conditional on that theorem being correctly cited. No internal flaw requiring a verdict change was found.","tokens_in":28430,"tokens_out":40047,"duration_ms":533165,"concrete_test":"Consult the original statements in Raghunathan [35] and Venkataramana [43] and verify that Theorem 6.2, exactly as cited, applies to every Γ=G(R) produced by Proposition 3.5 under Theorem 1.2. In particular, confirm that rank_F(G)≥1, that ∑_{v∈S} rank_{F_v}(G)≥2 for each such Γ, that G is simply connected, and that the ideal I is arbitrary nonzero. If a covered case has local rank sum 1 or fails any other hypothesis, the proof of Theorem 6.1 and therefore Theorem 1.2 is incomplete for that case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive step occurs at the end of the proof of Theorem 6.1: from a common nonzero vector in Im(p)∩Im(q), the argument needs the subgroup ⟨U(I_n),V(I_n)⟩ to have finite index in Γ=G(R), so that the cyclic Γ-space generated by the vector is finite-dimensional and contradicts weak mixing. This finite-index statement is Theorem 6.2, quoted from Raghunathan [35] and Venkataramana [43]. The paper explicitly says this is the only place where char(F)≠2 is used and that validity in characteristic 2 is unknown. Thus the central theorem is exactly as strong as Theorem 6.2, and the characteristic-two restriction is an honest boundary rather than an internal gap. One clarifying point worth recording: Proposition 3.5 hands us rank_F(G)=1 and |S|≥2, whereas Theorem 6.2's hypotheses as stated in Remark 6.3 involve the local rank sum ∑_{v∈S} rank_{F_v}(G)≥2. For the groups produced under Theorem 1.2 this should follow from rank(H)≥2, but it is the kind of hypothesis that must be checked against the original sources for every case, especially when some place in S is compact or contributes higher local rank. If Theorem 6.2 fails or its hypotheses are not met in some covered case, the contradiction in Theorem 6.1 collapses, and neither proof of Theorem 1.2 has a replacement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves character rigidity for all irreducible non-uniform lattices in semisimple groups of rank at least two, under the assumptions that the ambient local fields have characteristic different from 2 and, in the case of Theorem 1.2, that the semisimple group admits an almost simple rank-one factor. The proof reduces to arithmetic lattices Γ = G(R) via Margulis' arithmeticity theorem, then proves a new mixing theorem (Theorem 6.1): every weakly mixing trace on Γ restricts to a mixing trace on an F-split torus, with uniform decay along elements of the form a^i u. The key ingredients are a detailed character theory of solvable arithmetic groups (§5, congruence characters and induced characters), the Raghunathan–Venkataramana finite-index generation theorem (Theorem 6.2), and a careful Bruhat-decomposition argument in rank one. Two independent proofs of Theorem 1.2 are given: one self-contained using the generalized Bekka lemma, and one using charmenability and reduction to the Peterson–Thom theorem for SL2. The paper also derives stabilizer rigidity for probability-measure-preserving actions (Corollary 1.4), invariant random subgroup rigidity, and an extension to lattices with infinite center (Theorem 1.5).","tokens_in":28671,"tokens_out":22217,"duration_ms":288025,"significance":"If the main theorem is correct, this is a substantial advance on the Stuck–Zimmer conjecture: it removes, for all non-uniform higher-rank lattices in characteristic different from 2, the previous dependence on a Kazhdan property-(T) factor. The paper includes two independent proofs of the central result, which strengthens confidence in the argument, and it treats the non-property-(T) case (products of rank-one factors) that was the main open territory. The dependence on the external Raghunathan–Venkataramana theorem is stated honestly, including the explicit caveat that characteristic 2 is excluded precisely because that theorem is not known there. The proof is parameter-free and has no fitted constants or normalization tricks; the main line is structurally coherent and the Berkovich-style reductions through commensurability and central extensions are handled abstractly. The derived applications to stabilizers and invariant random subgroups are natural and clearly explained.","major_comments":[{"comment":"The proof of Theorem 6.1 invokes Theorem 6.2 at the final step, where the subgroups U_n and V_n generate a finite-index subgroup of Γ = G(R). Theorem 6.2 requires, according to Remark 6.3, that ∑_{v∈S} rank_{F_v}(G) ≥ 2. This condition is not part of the Standing Assumptions of §3.1, which only require rank_F(G) = 1, |S| ≥ 2, and char(F) ≠ 2. In the applications obtained through Proposition 3.5 the condition does follow from rank(H) ≥ 2, because at least two local places contribute noncompact factors; however, as Theorem 6.1 is stated, it covers cases where the hypothesis of Theorem 6.2 is not known to hold. The mismatch is load-bearing, since the finite-index conclusion is precisely what produces the finite-dimensional subrepresentation contradicting weak mixing. The authors should either add the local-rank-sum condition to the Standing Assumptions/Theorem 6.1 or explicitly verify it before the invocation of Theorem 6.2.","section":"§6, proof of Theorem 6.1"},{"comment":"The proof of Lemma 8.4 contains an invalid inference: it asserts that because Γ is finitely generated and separated by its finite-dimensional unitary representations, it is residually finite. This does not follow, since a finite-dimensional unitary representation of a finitely generated group need not have finite image. Separation by such representations therefore does not yield finite quotients separating points. Lemma 8.4 is used in Lemma 8.5 to establish that the lattice Λ in Theorem 1.5 has at most countably many finite-dimensional unitary representations, and hence Theorem 1.5 depends on it. This does not affect the main Theorem 1.2, whose proof uses [4, Proposition 7.1] directly, but it is a genuine gap in the statement of Theorem 1.5 and should be repaired with a valid argument or by restricting the scope of the lemma.","section":"§8.3, Lemma 8.4"}],"minor_comments":[{"comment":"The proof says that the sequence (p∧q)∨p_n − p_n converges to 0 in the strong operator topology, but what is actually established is trace convergence. Since the subsequent argument only uses trace convergence and the normality of τ, the statement and proof should be adjusted accordingly.","section":"§6, Lemma 6.7"},{"comment":"The proof is compressed: the verification that H_mix is a closed invariant subspace and the uniqueness of the decomposition are left to the reader. These facts are straightforward, but given that the lemma is used later for the restriction to a torus, a few lines of detail would improve readability.","section":"§2.3, Lemma 2.6"},{"comment":"The proof is omitted with a reference to a 'straightforward verification' by induction and restriction. Since the lemma is used in Proposition 4.2, a short proof or an explicit reference would be preferable.","section":"§8.3, Lemma 8.3"},{"comment":"In the case where H has no factor with Kazhdan's property (T), the proof silently reduces to Theorem 1.2, which requires an almost simple rank-one factor. The authors should state the standard fact they are using: a simple factor over a local field without property (T) has rank one, so the hypothesis of Theorem 1.2 is indeed satisfied.","section":"§8.1, proof of Theorem 1.3"},{"comment":"There are several typographical errors, e.g. 'Apriori' in §3, 'assocated' and 'combitation' in Proposition 5.8, and 'Induction' for 'inductions' in Lemma 8.3. These should be corrected in the final version.","section":"Throughout"},{"comment":"The notation ψ_i(·) = φ_i(π_{φ_i}(·)p_i) is ambiguous, since φ_i is first introduced as a function on the group. Clarifying the identification of the trace on the von Neumann algebra with the function on the group would help the reader.","section":"§2.2, proof of Proposition 2.4"}],"recommendation":"major_revision","confidential_remarks":"The two major issues are local and appear fixable: adding the local-rank-sum hypothesis to Theorem 6.1 or verifying it in the proof, and replacing the invalid residual-finiteness step in Lemma 8.4. The main theorem and the overall strategy are convincing; I do not see a circularity or a fundamental flaw. If these points are addressed, I would view the paper as acceptable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on 2507.21862. The paper is the real thing: it proves character rigidity for every non-uniform irreducible higher-rank lattice in semisimple groups of characteristic different from 2, which resolves the Stuck–Zimmer conjecture for that entire class. The main new engine is Theorem 6.1, a mixing theorem for weakly mixing traces on the F-split torus of an arithmetic rank-one group, and the paper gives two independent proofs of the main theorem, both running through that result. The congruence-character machinery in Section 5 is a genuine technical contribution, not a repackaging.\n\nThe authors are also honest about the boundary: the only place char(F)≠2 enters is Raghunathan–Venkataramana's theorem (Thm 6.2), and they say explicitly they don't know whether it holds in characteristic 2. That's exactly the right way to handle a restriction. The reduction in Proposition 3.5, showing that every lattice in the statement is commensurable (up to central extension) to a group G(R), is careful and makes the arithmetic setup credible.\n\nNow the soft spots, in proportion. The central theorem is exactly as strong as the quoted theorem of Raghunathan and Venkataramana. If that theorem fails, both proofs collapse and there is no fallback. That's a dependence, not a defect, but it deserves to be explicit. More concretely, Theorem 6.1 as stated invokes Theorem 6.2, whose hypothesis is sum of local ranks at least 2. The Standing Assumptions only give rank_F(G)=1 and |S|≥2, which is not enough in general (a quaternion group over Q split at exactly one place in S is a counterexample to the Standing Assumptions implying the sum). In the cases that matter for Theorem 1.2, the condition follows from rank(H)≥2 via Proposition 3.5, so it's fixable by adding a clause to the Standing Assumptions or to Theorem 6.1. A referee should chase that. Minor: Lemmas 2.6 and 8.3 are left to the reader (both are routine), and the proof of Lemma 6.7 claims SOT convergence where the argument really needs trace convergence; the trace statement is true and sufficient, so this is a small presentation slip.\n\nWho should read it: anyone working on characters, invariant random subgroups, or stabilizer rigidity for lattices. It deserves a serious referee. Send it out, but the referee should ask the authors to make the local-rank-sum hypothesis explicit wherever Theorem 6.2 is used. I'd take the result to a reading group and would cite it as the resolution of the non-uniform case.","headline":"A major step: character rigidity for all non-uniform higher-rank lattices in char ≠2, with an honest dependence on a deep external theorem.","tokens_in":29212,"tokens_out":6329,"would_cite":true,"duration_ms":69650,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","20C07","22D10","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every irreducible non-uniform higher-rank lattice is character rigid.","keywords":["character rigidity","irreducible lattices","non-uniform lattices","semisimple groups","Stuck–Zimmer conjecture","invariant random subgroups","congruence characters","mixing traces"],"falsifier":"Compute the subgroup generated by the two opposite congruence subgroups U(I) and V(I) inside G(R) for a rank-one isotropic group over a global field of characteristic 2: if some non-zero ideal I gives a proper subgroup of infinite index, the generation theorem fails and the main mixing argument breaks. Equivalently, produce a weakly mixing trace φ on G(R) with limsup of |φ(a^i u)| not tending to zero for a Zariski-dense a; this would refute Theorem 6.1 and with it the claimed character rigidity.","tokens_in":1976,"feed_emoji":"📐","tokens_out":6233,"duration_ms":143201,"temperature":0.7,"pith_summary":"The paper proves that every irreducible non-uniform lattice in a semisimple group of rank at least two is character rigid, provided the underlying local fields do not have characteristic 2. Character rigidity means any indecomposable trace — a normalized, conjugation-invariant positive-definite function — is either built from a finite-dimensional unitary representation or vanishes outside the center. The result closes the non-uniform case of the Stuck–Zimmer conjecture: every ergodic probability-measure-preserving action of such a lattice is either essentially transitive or has all non-central elements acting with fixed-point sets of measure zero. It also forces every ergodic invariant random subgroup to be either supported on the conjugacy class of a finite-index subgroup or concentrated on a central subgroup. The proof works by establishing a mixing phenomenon: any weakly mixing trace decays along elements of a split torus, uniformly over unipotent directions.","feed_headline":"All non-uniform higher-rank lattices are character rigid","feed_subtitle":"Every trace of these lattices is finite-dimensional or vanishes outside the center, settling Stuck–Zimmer.","key_machinery":"The load-bearing mechanism is a dichotomy for characters of the solvable arithmetic semidirect product $M = \\langle a \\rangle \\ltimes U(R)$, where $U$ is the unipotent radical of a minimal parabolic and $a$ generates a Zariski-dense subgroup of a split torus. Proposition 5.6 says each character of $M$ is either induced from $U$ (hence infinite-dimensional and vanishing off $U$) or restricts to a congruence trace on $U$ (hence finite-dimensional). Congruence characters are detected dynamically: a character of $U(R)$ is congruence exactly when its orbit under the dual action of $\\langle a \\rangle$ is finite (Proposition 5.5). Theorem 6.1 then uses the theorem that opposite congruence subgroups $U(I)$ and $V(I)$ together generate a finite-index subgroup of $\\Gamma$; the contradiction argument shows that a non-mixing component on the split torus would produce a common invariant vector for these two subgroups and hence a finite-dimensional subrepresentation, contradicting weak mixing.","core_discovery":"On the paper's own terms, the discovery is a decay theorem (Theorem 6.1) for traces of the arithmetic lattices $\\Gamma = G(R)$ that arise from a global field $F$ of characteristic $\\neq 2$, a rank-one isotropic algebraic group $G$, and a ring of $S$-integers $R$ with $|S| \\geq 2$. The theorem says that every weakly mixing trace $\\varphi$ of $\\Gamma$ satisfies $\\lim_{i \\to \\infty} \\sup_{u \\in U(R)} |\\varphi(a^i u)| = 0$ for an element $a$ generating a Zariski-dense subgroup of a split torus and for $U$ the unipotent radical of a minimal parabolic subgroup. From this, character rigidity of $\\Gamma$ is derived by two routes: a self-contained Bruhat-decomposition argument using a generalized Bekka vanishing lemma, and a softer argument through charmenability that reduces the problem to the known rigidity of $SL_2(R)$. Combined with the arithmeticity theorem and existing property-(T) results, this yields Theorem 1.3: all non-uniform higher-rank irreducible lattices in characteristic $\\neq 2$, and all such lattices with a property-(T) factor, are character rigid.","pith_inferences":["The characteristic-2 restriction appears to be an artifact of the opposite-congruence generation theorem; if that theorem holds over characteristic-2 fields, the same proof should remove the restriction entirely.","The uniform decay in Theorem 6.1 may admit explicit rates tied to Dirichlet units and the filtration by congruence subgroups, giving quantitative versions of character rigidity not stated in the paper.","The congruence/non-congruence dichotomy for characters of unipotent arithmetic groups suggests a similar dichotomy for invariant random subgroups, linking character rigidity to the congruence subgroup property.","A natural test case beyond the paper's scope would be SL_2 over rings of S-integers in characteristic 2, where the same character classification should either hold or produce a counterexample that pinpoints the failure of the generation theorem."],"forward_implications":["Every ergodic probability-measure-preserving action of such a lattice is either essentially transitive or satisfies $\\mu(\\mathrm{Fix}(g)) = 0$ for every non-central $g$ (Corollary 1.4).","Every ergodic invariant random subgroup of such a lattice is either uniform on the conjugacy class of a finite-index subgroup or Dirac on a central subgroup.","The lattices considered are charfinite, so their uniformly recurrent subgroups and unitary representations inherit the corresponding rigidity properties.","The same character rigidity and stabilizer rigidity conclusions hold for irreducible lattices in higher-rank semisimple Lie groups with arbitrary, possibly infinite, center (Theorem 1.5).","The non-uniform case of the Stuck–Zimmer conjecture is now settled for all characteristics other than 2."],"supporting_citations":[{"why":"Provides the arithmeticity theorem and structural facts that reduce arbitrary irreducible non-uniform lattices to the groups $\\Gamma = G(R)$.","marker":"[32]"},{"why":"Supplies the generation theorem for opposite congruence subgroups, which is the only characteristic-sensitive step in the proof of Theorem 6.1.","marker":"[35, 43]"},{"why":"Provides the SL_2(R) character rigidity result and the original mixing proposition that Theorem 6.1 generalizes.","marker":"[34]"},{"why":"Gives charmenability of the arithmetic lattice, used in the second proof to reduce von Neumann amenable characters to finite-dimensional ones.","marker":"[3]"},{"why":"Supplies charmenability and charfinite results for product-type arithmetic groups, and countability of finite-dimensional unitary representations.","marker":"[4]"},{"why":"Provides the generalized Bekka vanishing lemma used in the first proof of character rigidity.","marker":"[14]"},{"why":"Supplies the induction and character-theory tools for solvable groups used in the classification of characters of $T \\ltimes U$.","marker":"[29]"},{"why":"Gives the Dirichlet unit theorem, which produces the Zariski-dense element $a$ in the split torus.","marker":"[46]"}],"fun_headline_variants":["Character rigidity holds for every non-uniform higher-rank lattice","Stuck–Zimmer conjecture confirmed for non-uniform higher-rank lattices","Trace decay drives character rigidity in all non-uniform higher-rank lattices","Every non-uniform higher-rank lattice is character rigid"],"cache_read_input_tokens":31360,"weakest_assumption_plain":"The argument rests on a known generation theorem: the congruence subgroups attached to two opposite unipotent subgroups together generate a finite-index subgroup of the lattice; should that theorem fail, for instance in characteristic 2, the key mixing contradiction would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Character rigidity holds for every non-uniform higher-rank lattice","Stuck–Zimmer conjecture confirmed for non-uniform higher-rank lattices","Trace decay drives character rigidity in all non-uniform higher-rank lattices","Every non-uniform higher-rank lattice is character rigid"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000471,"raw_usage":{"total_tokens":2287,"prompt_tokens":831,"completion_tokens":1456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":1387}},"tokens_in":447,"tokens_out":1456,"duration_ms":11988,"temperature":1.0,"reasoning_tokens":1387,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:17:46.060011+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the subgroup generated by the two opposite congruence subgroups U(I) and V(I) inside G(R) for a rank-one isotropic group over a global field of characteristic 2: if some non-zero ideal I gives a proper subgroup of infinite index, the generation theorem fails and the main mixing argument breaks. Equivalently, produce a weakly mixing trace φ on G(R) with limsup of |φ(a^i u)| not tending to zero for a Zariski-dense a; this would refute Theorem 6.1 and with it the claimed character rigidity.","supporting_citations":[{"cited_title":"17, Springer-Verlag, Berlin, 1991","cited_arxiv_id":null,"evidence_quote":"Provides the arithmeticity theorem and structural facts that reduce arbitrary irreducible non-uniform lattices to the groups $\\Gamma = G(R)$."},{"cited_title":"Reine Angew","cited_arxiv_id":null,"evidence_quote":"Provides the SL_2(R) character rigidity result and the original mixing proposition that Theorem 6.1 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives charmenability of the arithmetic lattice, used in the second proof to reduce von Neumann amenable characters to finite-dimensional ones."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies charmenability and charfinite results for product-type arithmetic groups, and countability of finite-dimensional unitary representations."},{"cited_title":"3, 3181–3229","cited_arxiv_id":null,"evidence_quote":"Supplies the induction and character-theory tools for solvable groups used in the classification of characters of $T \\ltimes U$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Dirichlet unit theorem, which produces the Zariski-dense element $a$ in the split torus."}],"review_version":1}