{"id":"ddf65cb0-83fc-4b05-b9d3-5a42b4301cd1","arxiv_id":"2411.13858","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Zimmer's dimension conjecture is proved for many non-split semisimple Lie groups, including all complex semisimple groups without rank-1 factors.","lead":"This paper proves new cases of a famous rigidity conjecture about when large discrete subgroups of Lie groups can act on small manifolds. The proof introduces two techniques that should help settle the remaining cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.1's second subcase asserts γ=e1+e2 ∈ Σ_Q for Q with Π_Q={α2,...,α_n}; this inclusion is false, so the sharp SU(m,n), Sp(m,n), and SO*(2n) bounds are not established.","rationale":"The reader's weakest-assumption analysis pointed to the delegation in Lemma 5.6 and Proposition 5.8. I agree that this is a soft spot, but the most load-bearing concrete flaw occurs in its application, Proposition 6.1. The asserted root-space inclusion is checkable by root-system arithmetic and is false, so the contradiction argument in the x1=0 branch does not go through. This is not a stylistic dissatisfaction with 'standard arguments'; it is a definite incorrect statement. The paper is otherwise careful, and the surrounding entropy and superrigidity framework is plausible, so the flaw is potentially repairable: one would need to show directly that the non-atomic conditional measure along G[-e1] forces invariance under G_{-2e1} when its support lies in that subgroup. Because the error is local and may admit a fix, I would not reject the paper outright, but I also would not accept it in its current form. The appropriate verdict is CONDITIONAL: the proof of Theorem 1.3(iv) and (v), and Theorem 3.12, must be repaired in Proposition 6.1 before the claimed sharp thresholds are established.","tokens_in":37850,"tokens_out":26541,"duration_ms":235761,"concrete_test":"Check the claimed inclusion in Proposition 6.1 by computing Σ_Q for Q with Π_Q={α2,...,α_n} in SU(4,4) (type C_4): e1+e2=α1+2α2+2α3+α4, so e1+e2∉Σ_Q. This confirms the false step. Then decide the branch x1=0: prove directly from Lemma 5.6(1) that if µ^{G[-e1],E}_x is non-atomic then the invariant subgroup V is exp(g_{-2e1}) (or otherwise derive G_{-2e1}-invariance), and re-run Case 1 of (6.6) for, say, SU(7,4) and for the m=n cases. If the re-derived lower bound is still 2m+2n-3, the theorem survives; if not, the sharp thresholds in (iv) and (v) are unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 6, proof of Proposition 6.1, branch x1=0: the authors choose β=-2e1, γ=e1+e2, x=x2 and state 'In each case, we have γ∈Σ_Q.' For the standard parabolic Q with Π_Q={α2,...,α_n}, Σ_Q=Σ^+∪(span(Π_Q)∩Σ). In type C_n, e1+e2=α1+2α2+...+2α_{n-1}+α_n; in type BC_n the analogous expansion also has nonzero α1 coefficient. Hence e1+e2∉Σ_Q and g_{e1+e2}⊄Lie(Q). Consequently µ^{G[γ]}_x is not known to be Haar, and the application of Proposition 2.4 to conclude G_{β+γ}-invariance is invalid. This branch is unavoidable: for G=SU(n,n) or SO*(4n), g_{-e1}=0, so every nonzero x1+x2 in g[-e1] has x1=0. In the m>n cases in (6.6), it is at least possible that the non-atomic conditional measure is supported in G_{-2e1}. Proposition 6.1 is the source of the '+1' over n(g∆) used in Case 1 of (6.5) and (6.6); without it the stated lower bound dim(M)≥n(g∆)+1 is unsupported, and the r0 bound from Proposition 5.10 is strictly weaker in the large-m regimes of Theorem 1.3 (e.g., SU(7,4)). Thus the proof of Theorem 1.3(iv),(v) and Theorem 3.12 has a gap at a load-bearing point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves new cases of Zimmer's conjecture for actions of lattices in non-R-split semisimple Lie groups on compact manifolds. The main theorem (Theorem 1.3) establishes the conjectured sharp dimension thresholds for complex semisimple groups without rank-one factors, SL(n,H) for n at least 5, SO+(m,n) in certain ranges, SU(m,n), SO*(2n) for n at least 14, and EII; for Sp(m,n) a threshold of v(G)-2 is proved (Theorem 3.12), and a volume-preserving analogue is given (Theorem 1.6). The strategy reduces the conjecture, via Theorem 3.1 and Corollary 3.10, to proving that s(G)=v(G), where s(G) is the largest integer below which every minimal-parabolic-invariant measure is G-invariant. Two mechanisms are introduced: cocycle superrigidity applied to Levi factors (Proposition 4.1) and measure rigidity for one-dimensional fiberwise coarse Lyapunov distributions (Theorem 3.14 and Proposition 5.8). The remaining sections contain extensive root-system and representation-theoretic estimates yielding the s(G) bounds.","tokens_in":38205,"tokens_out":36907,"duration_ms":390113,"significance":"If correct, this is a substantial advance: it gives sharp Zimmer thresholds for several families of non-split groups, where previously only the weaker resonant-codimension bound r(G) was available. The paper is careful in isolating the quantity s(G), and the root-space computations are explicit and checkable. The representation classification in Lemma 6.3 and the combinatorial case analyses in Section 6 are concrete strengths. I specifically checked the potential gap in Proposition 6.1 flagged during the review process, and it does not land: for the standard parabolic Q with Pi_Q={alpha2,...,alpha_n}, the root e1+e2 is a positive root in type C_n or BC_n, so it belongs to Sigma^+ and hence to Sigma_Q by the definition in Section 2.1; the x1=0 branch of Proposition 6.1 is therefore valid. The main expositional risk is that the homogenity conclusion in Lemma 5.6 is delegated to 'standard arguments' from [19] and [20]; the surrounding proof is plausible and cites precise sources, but a fuller statement would increase confidence.","major_comments":[{"comment":"The suspected gap in the x1=0 subcase is not present. For the standard parabolic Q with Pi_Q={alpha2,...,alpha_n}, Section 2.1 defines Sigma_Q = Sigma^+ union (span(Pi_Q) intersect Sigma). The root e1+e2 is a positive root in type C_n or BC_n, so e1+e2 belongs to Sigma^+ and hence to Sigma_Q; the fact that its expansion in simple roots has a nonzero alpha1 coefficient is irrelevant. The subsequent applications of Lemma 6.2 and Proposition 2.4 are therefore legitimate. This removes the concern that the '+1' in equations (6.5) and (6.6) is unsupported.","section":"Section 6, proof of Proposition 6.1"}],"minor_comments":[{"comment":"In the proof of case (2), the sentence 'The natural projection p : D2 x R -> R is proper and is measurably one-to-one' is inaccurate as written because p is not globally injective; the argument only needs the restriction of p to K_x (or to the support of Psi_x) to be one-to-one, which is established earlier. Please clarify this wording.","section":"Section 5.4, Lemma 5.6"},{"comment":"The notation 'E10' in the sentence 'H^{p,kappa}(M,m,S^2T*M) is in E10' is undefined; please supply the intended symbol or a reference for this space.","section":"Section 3.1, proof of Theorem 3.2"},{"comment":"There are numerous typographical artifacts from the arXiv source, including '/integerdivide' for notin, '/greaterorequalslant' for >=, 'Rimannian' for Riemannian, and 'compat' for compact; these should be cleaned in the final version.","section":"Throughout the manuscript"},{"comment":"Reference [28] is listed without an arXiv identifier. Since the text states that this preprint will not be submitted for publication, please provide a stable identifier or remove reliance on it.","section":"References"},{"comment":"Proposition 5.8 applies Lemma 5.6(2) without explicitly verifying at that point that the leafwise measure mu^{W[beta],E_beta}_x is non-atomic; this follows from Lemma 5.1, but stating the verification would improve readability.","section":"Section 5.4, Proposition 5.8"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a strong contribution and I do not see a load-bearing flaw. The stress-test concern about Proposition 6.1 is based on a misreading of Sigma_Q, since positive roots are always contained in Sigma_Q. The main risk is that Lemma 5.6's homogenity step is summarized rather than fully carried out; if the editors wish, an additional expert check of that lemma would be prudent. I have no citation or novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this is the real thing—a serious, technically dense advance on Zimmer's conjecture—and the specific Section 6 objection in the stress-test note is wrong. In Proposition 6.1, branch x1=0, the paper takes β=-2e1, γ=e1+e2 and claims γ∈Σ_Q. For Q with Π_Q={α2,...,αn}, one has Σ_Q=Σ^+∪(span(Π_Q)∩Σ). The root e1+e2 is positive, so it lies in Σ_Q regardless of its α1-coefficient. The stress-test forgot the Σ^+ term. Hence the conditional measure along G[γ] is Haar, and the rest of that branch is coherent. That particular gap is not there.\n\nWhat is actually new: two proof techniques—cocycle superrigidity for Levi factors and a one-dimensional Lyapunov dichotomy—and they deliver sharp v(G) bounds, not just the weaker r(G) bounds, for complex semisimple groups without rank-1 factors, SL(n,H), certain SO+(m,n), SU(m,n), SO*(2n), and EII. The volume-preserving boundary result is also new. The combination of the two approaches in Section 6 is genuinely clever and does not reduce the target to an input.\n\nWhere I am more cautious: the measure-rigidity core in Lemma 5.6 is delegated to 'standard arguments' from Kalinin–Spatzier and Katok–Spatzier. That homogenity conclusion is load-bearing for Proposition 5.8, and a referee should ask for the details to be written out. This is a real soft spot, but not obviously fatal; the surrounding structure and the entropy equation (5.2) are careful. Lemma 6.3, the uniqueness of the minimal representation, is also terse—again worth a referee's attention, but the cited sources point the right way.\n\nThis paper deserves a serious referee. I would send it to an expert in smooth rigidity, maybe two. It is not a reshuffling of known results; it advances the sharp threshold for several families that were previously open. If the delegated measure-rigidity step checks out, this is a clean resolution for the listed non-split cases. I would bring it to reading group if the group has the background, and I would cite it if I worked in the Zimmer program.","headline":"Serious, technically deep advance on Zimmer's conjecture; the stress-test's Section 6 objection doesn't survive contact with the definition of Σ_Q.","tokens_in":38747,"tokens_out":3134,"would_cite":true,"duration_ms":29097,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C85","37D25","22E40","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves Zimmer's conjecture for many non-split semisimple Lie groups, showing that lattice actions on compact manifolds below a sharp dimension threshold must preserve a continuous Riemannian metric.","keywords":["Zimmer's conjecture","lattice actions","non-split semisimple Lie groups","measure rigidity","Lyapunov exponents","cocycle superrigidity","parabolic subgroups","invariant metrics"],"falsifier":"Exhibit, for one of the listed groups, a $P$-invariant $P$-ergodic measure on a suspension space with $\\dim E^{[\\beta]}_F = 1$ and $\\dim \\mathfrak{g}^{[\\beta]} \\ge 2$ that is not $G^{[\\beta]}$-invariant; this would refute Proposition 5.8 and break the dimension bound $r_0(G)$. Equivalently, construct a $C^{1+\\kappa}$ action of a lattice on a compact manifold of dimension below $v(G)$ without a continuous invariant Riemannian metric.","tokens_in":37644,"feed_emoji":"📐","tokens_out":8768,"duration_ms":965139,"temperature":0.7,"pith_summary":"This paper proves many new cases of Zimmer's conjecture for actions by lattices in non-$\\mathbb{R}$-split semisimple Lie groups: for complex semisimple groups without rank-one factors, $\\mathrm{SL}(n,\\mathbb{H})$ with $n \\ge 5$, certain $\\mathrm{SO}^+(m,n)$, $\\mathrm{SU}(m,n)$, $\\mathrm{SO}^*(2n)$, and the exceptional group $\\mathrm{EII}$, any $C^{1+\\kappa}$ action on a compact manifold of dimension below a sharp threshold $v(G)$ must preserve a continuous Riemannian metric, and actions on still smaller manifolds have finite image. The proof reduces the conjecture to the study of probability measures invariant under a minimal parabolic subgroup, then introduces two mechanisms to force extra invariance of such measures: cocycle superrigidity applied to Levi factors of parabolic stabilizers, and a measure-rigidity dichotomy for one-dimensional fiberwise coarse Lyapunov distributions. If the dichotomy holds, these non-split groups satisfy the same sharp dimension bounds as their split counterparts, so the nonlinear rigidity phenomenon is not confined to $\\mathbb{R}$-split groups.","feed_headline":"Zimmer's conjecture proved for many non-split Lie groups","feed_subtitle":"Actions on manifolds below a sharp dimension threshold must preserve a metric or have finite image.","key_machinery":"The central object is the fiberwise coarse Lyapunov distribution $E^{[\\beta]}_F$: the subspace of the fiber tangent bundle in the suspension space whose vectors grow at a rate proportional to a restricted root $\\beta$ under the $A$-action. The new measure-rigidity step is Proposition 5.8, which says that when this distribution is one-dimensional, either the measure is invariant under the root subgroup $G^{[\\beta]} = \\exp(\\mathfrak{g}^{[\\beta]})$, or the root space $\\mathfrak{g}^{[\\beta]}$ itself has dimension one. This dichotomy, together with Zimmer's cocycle superrigidity theorem applied to Levi factors of parabolic stabilizers, turns the absence of an invariant metric into a quantitative lower bound on the dimension of the manifold.","core_discovery":"The central claim is Theorem 1.3: for each listed non-split group $G$, every $C^{1+\\kappa}$ action of a lattice $\\Gamma$ on a compact manifold $M$ with $\\dim(M) < v(G)$ preserves a continuous Riemannian metric, and if $\\dim(M) < \\min\\{v(G), v(G_{\\mathrm{cpt}})\\}$ then the image $\\alpha(\\Gamma)$ is finite. The proof's engine is the measure-rigidity dichotomy in Proposition 5.8: for a $P$-invariant ergodic measure whose fiberwise coarse Lyapunov distribution $E^{[\\beta]}_F$ is one-dimensional, either the measure is already $G^{[\\beta]}$-invariant or the root space $\\mathfrak{g}^{[\\beta]}$ is one-dimensional. Combining this dichotomy with cocycle superrigidity gives the lower bound $\\dim(M) \\ge r_0(G)$, and for the groups in Theorem 1.3 this bound equals $v(G)$, the conjectured sharp threshold.","pith_inferences":["If the dichotomy of Proposition 5.8 can be established for the remaining root-space configurations, the $\\mathrm{Sp}(m,n)$ gap of $2$ should close, giving the full conjecture for that family as well.","The proof suggests that the equality $r_0(G) = v(G)$ is the right general criterion for non-split groups; a natural test is to compute $r_0$ for the exceptional groups $\\mathrm{F}_4$, $\\mathrm{E}_6$, and $\\mathrm{E}_7$.","Because Lemma 5.6 delegates a key homogenity conclusion to 'standard arguments' in earlier work, the most direct check of the new engine is to write out that step in full for a concrete case such as $\\mathrm{SU}(3,3)$."],"forward_implications":["For each group in Theorem 1.3, $s(G) = v(G)$, so the non-volume-preserving Zimmer conjecture holds with the sharp threshold $v(G)$.","For $\\mathrm{Sp}(m,n)$ with $6 \\le n \\le m \\le \\frac14(n^2-3n+6)$, the gap to the conjecture is at most $2$: actions on manifolds of dimension below $v(G)-2$ have finite image.","In the volume-preserving case at the boundary $\\dim(M) = v(G)$, the action preserves a continuous Riemannian metric for complex semisimple groups without rank-one factors, $\\mathrm{SO}^+(n,n+2)$, $\\mathrm{SU}(n,n)$, and $\\mathrm{EII}$.","The bound $r_0(Q)$ in Proposition 5.10 is a new general dimension estimate that supersedes the earlier minimal-resonant-codimension bound $r(G)$ in the cases covered."],"supporting_citations":[{"why":"Reduces Zimmer's conjecture to subexponential derivative growth and provides the strong-property-(T) argument that turns this growth into an invariant metric.","marker":"[4]"},{"why":"Supplies the suspension-space construction, escape-of-mass estimates, and the reduction of invariant metrics to vanishing fiberwise Lyapunov exponents for non-uniform lattices.","marker":"[6]"},{"why":"Provides Theorem 3.6, which locates root spaces outside the fiberwise Lyapunov spectrum inside the stabilizer of the measure, and the coarse Lyapunov foliation framework.","marker":"[7]"},{"why":"Gives the conditional-measure machinery along invariant foliations and Proposition 2.4 on invariance from bracket combinations of support vectors.","marker":"[12]"},{"why":"Provides the 'standard arguments' for homogenity of leafwise measures that Lemma 5.6 invokes in the measure-rigidity dichotomy.","marker":"[19]"},{"why":"Supplies the corrected homogenity arguments for invariant measures of higher-rank abelian actions, also cited in Lemma 5.6.","marker":"[20]"},{"why":"Yields the Ledrappier-Young entropy formula used in Lemma 2.5 and Corollary 5.2 to bound entropy along coarse Lyapunov manifolds.","marker":"[22]"},{"why":"Provides the existence of ergodic elements used to select $a_0 \\in A'_\\beta$ in the proof of Lemma 5.6.","marker":"[25]"},{"why":"States Zimmer's cocycle superrigidity theorem, the engine of the superrigidity approach and the source of dimension bounds from Levi factors.","marker":"[29]"}],"fun_headline_variants":["Zimmer's conjecture: new non-split cases proved","Many non-split Lie groups yield to Zimmer's conjecture","Zimmer's conjecture verified for new non-split families","Non-split groups: Zimmer's conjecture dimension threshold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole new bound rests on the claim that whenever a fiberwise instability direction in the suspension space is one-dimensional, the measure must either acquire extra symmetry or the corresponding root space is one-dimensional; if that claim fails, the sharp thresholds do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Zimmer's conjecture: new non-split cases proved","Many non-split Lie groups yield to Zimmer's conjecture","Zimmer's conjecture verified for new non-split families","Non-split groups: Zimmer's conjecture dimension threshold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000771,"raw_usage":{"total_tokens":3381,"prompt_tokens":879,"completion_tokens":2502,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":2436}},"tokens_in":495,"tokens_out":2502,"duration_ms":18641,"temperature":1.0,"reasoning_tokens":2436,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:48:50.417120+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit, for one of the listed groups, a $P$-invariant $P$-ergodic measure on a suspension space with $\\dim E^{[\\beta]}_F = 1$ and $\\dim \\mathfrak{g}^{[\\beta]} \\ge 2$ that is not $G^{[\\beta]}$-invariant; this would refute Proposition 5.8 and break the dimension bound $r_0(G)$. Equivalently, construct a $C^{1+\\kappa}$ action of a lattice on a compact manifold of dimension below $v(G)$ without a continuous invariant Riemannian metric.","supporting_citations":[{"cited_title":"Brown, D","cited_arxiv_id":null,"evidence_quote":"Reduces Zimmer's conjecture to subexponential derivative growth and provides the strong-property-(T) argument that turns this growth into an invariant metric."},{"cited_title":"Brown, F","cited_arxiv_id":null,"evidence_quote":"Provides Theorem 3.6, which locates root spaces outside the fiberwise Lyapunov spectrum inside the stabilizer of the measure, and the coarse Lyapunov foliation framework."},{"cited_title":"Einsiedler, A","cited_arxiv_id":null,"evidence_quote":"Gives the conditional-measure machinery along invariant foliations and Proposition 2.4 on invariance from bracket combinations of support vectors."},{"cited_title":"Kalinin and R","cited_arxiv_id":null,"evidence_quote":"Provides the 'standard arguments' for homogenity of leafwise measures that Lemma 5.6 invokes in the measure-rigidity dichotomy."},{"cited_title":"Katok and R","cited_arxiv_id":null,"evidence_quote":"Supplies the corrected homogenity arguments for invariant measures of higher-rank abelian actions, also cited in Lemma 5.6."},{"cited_title":"Ledrappier, L.-S","cited_arxiv_id":null,"evidence_quote":"Yields the Ledrappier-Young entropy formula used in Lemma 2.5 and Corollary 5.2 to bound entropy along coarse Lyapunov manifolds."},{"cited_title":"Pugh and M","cited_arxiv_id":null,"evidence_quote":"Provides the existence of ergodic elements used to select $a_0 \\in A'_\\beta$ in the proof of Lemma 5.6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States Zimmer's cocycle superrigidity theorem, the engine of the superrigidity approach and the source of dimension bounds from Levi factors."}],"review_version":1}