{"id":"3ddc8be5-9769-4ad4-ab12-bf25012bd62e","arxiv_id":"2606.01459","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Coamenable normal subgroups of Borel-Anosov subgroups preserve the Riemannian critical exponent but not the full limit cone or growth indicator; rigidity holds exactly on the opposition-invariant directions, and counterexamples show this is sharp.","lead":"This paper shows that coamenable normal subgroups of higher-rank Lie group actions do not always inherit the ambient group's directional growth rates, even when the quotient is the integers. It then pins down the exact part of the growth data that does survive: the scalar growth rate always survives, and the full directional spectrum survives precisely on a set of symmetric directions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5 asserts without proof that a coamenable normal subgroup of a Zariski-dense Borel-Anosov group is Zariski dense; this premise is load-bearing for Theorem 1.4 and is not established.","rationale":"I focused on the unproved coamenable-N-is-Zariski-dense assertion rather than the Proposition 4.7/4.8 estimates because the estimates are quoted from the literature and the C>0 extension is plausibly routine, whereas the Zariski-density assertion is used to justify the very first step of the main rigidity theorem (Theorem 1.4). If the premise were false, the conclusion could fail on i-fixed boundary directions and the proof would not go through. The concern is real but likely addressable: a short algebraic argument using the fact that Zar(N) is normal in G, together with the Tits alternative, should settle it. The reader's CONDITIONAL verdict is appropriate: not a fatal flaw, but a missing proof that should be supplied in revision. The reader's weakest_assumption listed the same assertion as a separate fragile premise, so we partially agree; the main emphasis of the reader was on the coarse-geometric estimates, while I consider the Zariski-density assertion to be the single most load-bearing gap.","tokens_in":32591,"tokens_out":31150,"duration_ms":307131,"concrete_test":"Write out a proof of the §5 assertion: for H = Zar(N), show H is normal in G; if H ≠ G, the image Δ of Γ in G/H is a Zariski-dense amenable subgroup. Check whether such a Δ can exist when Γ is Borel-Anosov, distinguishing noncompact G/H (contradiction via the Tits alternative) from compact G/H (where a dense amenable subgroup of a compact semisimple Lie group must be virtually abelian, hence cannot be Zariski dense). If the proof succeeds, insert it as a lemma; if it fails, construct the graph example Γ = {(g, ρ(g))} in SL2(R)×SL2(R) with N = {(g, ρ(g)) : g ∈ ker(Γ0→Z)} and compute Zar(N) to see whether it is proper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"At the start of §5 the authors state: “let N⊲Γ be a coamenable normal subgroup, which is necessarily Zariski dense.” No proof or reference is provided. This assertion is used critically in the proof of Theorem 5.2 (= Theorem 1.4). First, it is needed to apply Lemma 4.2, which gives L_N^i = L_Γ^i; without this equality the proof cannot even start, since the theorem asserts equality on i-fixed directions that could lie in L_Γ but not in L_N. Second, the assertion is used to guarantee that L_N has non-empty interior, which supplies the i-fixed interior point w used in the boundary-interpolation argument. The claim is not formal: from normality, Zar(N) is normalized by Γ and hence is a normal algebraic subgroup of G, but it could a priori be a proper product of simple factors. Coamenability of Γ/N only forces the image of Γ in G/Zar(N) to be an amenable Zariski-dense subgroup; ruling this out requires an argument (Tits alternative plus a careful treatment of compact quotients), which the paper does not supply. If the premise failed, L_N^i could be strictly smaller than L_Γ^i and Theorem 1.4 would collapse. The reader flagged this separately, but it is the most load-bearing gap because it is an unproved structural premise rather than a quoted estimate that could be checked against the literature.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies higher-rank analogues of Roblin's theorem on coamenable normal subgroups of discrete groups. For Zariski-dense Borel-Anosov subgroups of connected semisimple real algebraic groups, the authors prove that the Riemannian critical exponent is preserved under coamenable normal subgroups (Theorem 1.3), and that the growth indicator is preserved exactly on the fixed-point set of the opposition involution (Theorem 1.4). They also show that full directional rigidity fails: for every odd n≥3 there is an open family of Zariski-dense Borel-Anosov Schottky subgroups of SL_n(R) with cocyclic normal subgroups having strictly smaller limit cones (Theorem 1.1), and in SL_3(R) there are cocyclic pairs with equal limit cones but different growth indicators at an interior direction (Theorem 1.5). The main technical inputs are a perturbative control of Jordan projections for free groups (Propositions 2.2–2.3) and a coamenability theorem for dual critical exponents (Theorem 4.5), proved by weighted Poincaré series and amenable averaging over Γ/N. The appendix proves a general lower bound δ_N ≥ δ_Γ/2 for arbitrary infinite normal subgroups.","tokens_in":32901,"tokens_out":12845,"duration_ms":116673,"significance":"If the few missing justifications are supplied, these results provide a complete and sharp higher-rank replacement for Roblin's rank-one theorem. The counterexamples are explicit, robust under perturbation, and the positive rigidity results are proved by a flexible averaging argument that is likely to extend to θ-Anosov and relatively Morse settings, as noted in Remark 4.12. The appendix's general lower bound δ_N ≥ δ_Γ/2 is also a useful standalone contribution. The paper is careful and cites the relevant literature; it does not rely on parameter fitting or circular reasoning. These are significant advances in the study of growth of higher-rank discrete subgroups.","major_comments":[{"comment":"The assertion that a coamenable normal subgroup N of a Zariski-dense Borel-Anosov subgroup Γ is 'necessarily Zariski dense' is unproved and load-bearing. It is used to apply Lemma 4.2 (to obtain L_N^i = L_Γ^i) and to ensure that L_N has non-empty interior, which is essential for the boundary-interpolation argument in Theorem 5.2. Without this premise Theorem 1.4 does not follow. A proof or precise reference is needed. A plausible argument: let H=Zar(N); then H is a normal algebraic subgroup of G, and the image of Γ in G/H is a quotient of Γ/N, hence amenable, while also being Zariski dense in G/H. Since amenable linear groups are virtually solvable and have solvable Zariski closure, G/H must be trivial. The authors should spell this out, including the treatment of finite centers and compact factors.","section":"§5, first paragraph"},{"comment":"The extension of [23, Cor. 5.15] from the stated case C=0 to arbitrary C≥0 is asserted with the phrase 'its proof works for a general C>0' but no proof is given. If only the C=0 case is needed in the sequel (as appears to be the case for (4.7) along word geodesics), please state that and remove the unproved generalization; otherwise provide a proof. This is a missing justification in the foundational estimates used for Theorem 4.5.","section":"§4, Proposition 4.7"}],"minor_comments":[{"comment":"Typo: 'W will use' should be 'We will use'.","section":"§4, before Lemma 4.4"},{"comment":"The notation O_R(1) is used in the proof without defining it; it presumably means a constant depending only on R. Please clarify to avoid confusion.","section":"§4, Lemma 4.9"},{"comment":"The proof is somewhat compressed. In particular, the statement that 'by the convergence dynamics of the Anosov subgroup ∆, all but finitely many elements g∈∆ satisfy gX⊂intY' should be justified in more detail, and the simultaneous choice of ∆' satisfying both the ping-pong condition and the Zariski-density condition (2) needs a clearer argument. The lemma is important for the L_N=L_Γ refinement in Theorem 6.9.","section":"§6, Lemma 6.10"},{"comment":"The Borel-Anosov condition for the representation ρ_κ is quoted from [22]. Since [22] is a preprint, please ensure the precise statement and hypotheses are publicly available and that the condition (3.3) indeed matches [22].","section":"§3, equation (3.3)"},{"comment":"The statement 'L_Γ − {0} ⊂ int a^+' is attributed to [15]; it may depend on the Borel-Anosov assumption and on a choice of positive Weyl chamber. Please make the hypotheses explicit.","section":"§5, Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"This is a strong and important paper. The main gap is the unproved Zariski-density assertion for coamenable normal subgroups in §5; if the authors supply a proof or a precise reference, the paper should be acceptable. The constructions are explicit and the positive results are based on a robust averaging method. I would not reject. The other issues are local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the quick take. This paper gives the right higher-rank replacement for Roblin's theorem: coamenability does not preserve limit cones or the full growth indicator, but it does preserve the Riemannian critical exponent and the growth indicator on the opposition-invariant directions. The counterexamples are new and explicit, and the positive results are likely correct, but the written proof has one load-bearing gap that needs patching.\n\nWhat is actually new: Theorem 1.1 builds an open family of Zariski dense Borel-Anosov Schottky subgroups of SL_n(R) with cocyclic normal subgroups having strictly smaller limit cones, for all odd n≥3. The mechanism — twisting a 2d-dimensional irreducible representation by a character that records χ on the middle Jordan coordinate — is clean, and the perturbation estimate (Prop 2.3) is strong enough to give openness. Theorem 1.5, with equal limit cones but different growth indicators at an interior direction in SL_3, is a nice sharpness example; the pressure perturbation in Lemma 6.1 is convincing. The coamenability averaging in Section 4, leading to the symmetrized form φ̄, is the right tool, and the proof of Theorem 4.5 is careful. I also like the appendix bound δ_N ≥ δ_Γ/2 for arbitrary infinite normal subgroups.\n\nWhere I have concerns, in order of importance. First, Section 5 asserts that a coamenable normal subgroup of a Zariski dense Borel-Anosov group is necessarily Zariski dense, with no proof or reference. This is load-bearing: it is used to apply Lemma 4.2 (L_N^i = L_Γ^i) and to get an i-fixed interior point in L_N. The assertion is not formal; it needs a Tits-alternative argument that the image of Γ in G/Zar(N) cannot be both amenable and Zariski dense unless the quotient is compact. If G has compact factors, the statement as written is at least questionable. The fix is probably to assume G has no compact factors (or to work with the projection to the noncompact part), and then prove the claim. As written, the proof of Theorem 1.4 has a gap.\n\nSecond, Proposition 4.7 extends a cited estimate from C=0 to C>0 with no proof. This is likely harmless, but it should be checked. Third, Lemma 6.10, used to arrange L_N = L_Γ, is sketched rather than proved; the sketch is plausible but needs details.\n\nOverall, the main theorems are probably true, the counterexamples are solid, and the paper is an important contribution. It deserves a serious referee. I would send it with a request to fix the Zariski-density issue and fill the smaller gaps.","headline":"A strong paper that gives the clean higher-rank counterpart to Roblin's theorem; the counterexamples are new and the rigidity results are likely right, but the proof has a load-bearing unproved Zariski-density claim that needs fixing.","tokens_in":33420,"tokens_out":15396,"would_cite":true,"duration_ms":149711,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","20F65","20F67","53C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Coamenable subgroups preserve higher-rank growth exactly on opposition-symmetric directions.","keywords":["coamenable subgroup","Borel-Anosov subgroup","limit cone","growth indicator","critical exponent","opposition involution","Schottky subgroup","Roblin's theorem"],"falsifier":"Exhibit a single Zariski-dense Borel-Anosov pair (Γ,N), with N coamenable, where ψ_N(v) < ψ_Γ(v) for some v in int L_N with i(v)=v; or, at the level of the engine, exhibit a Borel-Anosov Γ and positive φ for which the coarse triangle inequality d_φ(x,z) ≤ d_φ(x,y)+d_φ(y,z)+D fails — in either case Theorem 1.4 and Theorem 4.5 collapse.","tokens_in":32459,"feed_emoji":"📐","tokens_out":6427,"duration_ms":54093,"temperature":0.7,"pith_summary":"Roblin's rank-one theorem says that a coamenable normal subgroup has the same critical exponent as its ambient group. The paper shows that the higher-rank directional analogue fails: for every odd n≥3 there are open families of Zariski-dense Borel-Anosov Schottky subgroups of SL_n(R) with infinite-cyclic normal subgroups whose limit cones, and hence growth indicators, are strictly smaller. Even when the limit cones coincide (in SL_3(R)), the growth indicator can differ at an interior direction. The rigidity that survives is precise: the Riemannian critical exponent is always preserved, and the growth indicator is preserved exactly on the fixed set of the opposition involution; when that involution is trivial, the full growth indicator is preserved. These three statements together are the correct higher-rank replacement for Roblin's theorem.","feed_headline":"Higher-rank Roblin rigidity fails except along symmetric directions","feed_subtitle":"Cocyclic normal subgroups can shrink limit cones; a new theorem pins down which growth directions must persist.","key_machinery":"The load-bearing object is the opposition involution i on the positive Weyl chamber, defined by µ(g^{-1}) = i(µ(g)); its fixed-point set is exactly where rigidity survives. The main technical engine is Theorem 4.5: the symmetrized inequality δ_{Γ,φ̄} ≤ δ_{N,φ} ≤ δ_{Γ,φ}, proved by defining a coarse pseudo-distance d_φ(γ_1,γ_2)=φ(µ(γ_1^{-1}γ_2)), using shadow estimates and coarse additivity of the Cartan projection for Anosov groups, and averaging a bounded logarithmic distortion over a right-invariant mean on Γ/N to obtain a character that cancels when the φ and φ∘i estimates are combined. For the counterexamples, the machinery is a perturbative ping-pong estimate (Propositions 2.2 and 2.3)","core_discovery":"The paper's central discovery is a symmetrized coamenability inequality controlling directional critical exponents. For a Zariski-dense Borel-Anosov subgroup Γ of a connected semisimple real algebraic group and a coamenable normal subgroup N, for every linear form φ positive on L_Γ∖{0}, the paper proves δ_{Γ,φ̄} ≤ δ_{N,φ} ≤ δ_{Γ,φ}, where φ̄ = (φ+φ∘i)/2 and i is the opposition involution (the involution satisfying µ(g^{-1})=i(µ(g))). Symmetric forms therefore give equality δ_{N,φ}=δ_{Γ,φ}. From this inequality the paper derives δ_N=δ_Γ and ψ_N=ψ_Γ on the i-fixed points of a^+, using convex duality and strict concavity of growth indicators. The paper also proves this is sharp: Theorem 1.1 pro","pith_inferences":["The symmetrized inequality suggests a general heuristic the paper leaves implicit: coamenability rigidity in higher rank is governed by the opposition involution, so one should expect the i-fixed locus to be the universal region where any directional invariant (limit cone, growth indicator, critical exponents of symmetric forms) is forced to coincide.","The perturbation estimate used for the counterexamples is open in the representation variety, so the failure of directional rigidity is not a single example but a stable, non-empty open phenomenon for Schottky-type representations in SL_n.","A testable extension: the proof of Theorem 4.5 relies only on coarse shadow estimates and coarse additivity, which are known for relatively Morse subgroups; if those estimates hold in that wider setting, the same rigidity should hold for cusped Hitchin or relatively Anosov subgroups — a direction the paper only remarks on.","The appendix's bound δ_N ≥ δ_Γ/2 for arbitrary infinite normal subgroups, together with the coamenable equality, suggests a quantitative hierarchy in which the exact constant measures how 'large' the normal subgroup is inside Γ; coamenability is the condition that upgrades the constant from 1/2 to 1."],"forward_implications":["If the opposition involution is trivial (no simple factors of type A_n, D_{2n+1}, E_6), then every coamenable normal subgroup of a Zariski-dense Borel-Anosov group has the full same growth indicator and the same critical exponent as the ambient group.","In rank one, where i is trivial, Roblin's theorem is recovered as the special case of Theorem 1.4.","Amenability of the quotient does not force the limit cone to be preserved: even an infinite cyclic quotient can have strictly smaller limit cone, stably under small deformations of the representation.","Equal limit cones do not force equal directional growth: in SL_3(R) there are cocyclic pairs with L_N = L_Γ but ψ_N ≠ ψ_Γ at an interior direction.","The Riemannian critical exponent is rigid for all coamenable normal subgroups of Zariski-dense Borel-Anosov groups, a genuinely higher-rank statement with no directional caveat."],"fun_headline_variants":["Coamenable subgroups can shrink limit cones in higher rank","Roblin's theorem fails in higher rank, but only off symmetric axes","Opposition-invariant directions force equal growth indicators","Cocyclic subgroups break limit-cone rigidity, except along involution axes"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof assumes that the coarse shadow and triangle estimates for the Cartan-projection pseudo-distance hold for Borel-Anosov subgroups and that a coamenable normal subgroup is necessarily Zariski dense; if either assumption fails, the lower bound δ_{N,φ} ≥ δ_{Γ,φ̄} and the rigidity theorems do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Coamenable subgroups can shrink limit cones in higher rank","Roblin's theorem fails in higher rank, but only off symmetric axes","Opposition-invariant directions force equal growth indicators","Cocyclic subgroups break limit-cone rigidity, except along involution axes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000592,"raw_usage":{"total_tokens":2643,"prompt_tokens":807,"completion_tokens":1836,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1776}},"tokens_in":551,"tokens_out":1836,"duration_ms":13179,"temperature":1.0,"reasoning_tokens":1776,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T12:37:20.249941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a single Zariski-dense Borel-Anosov pair (Γ,N), with N coamenable, where ψ_N(v) < ψ_Γ(v) for some v in int L_N with i(v)=v; or, at the level of the engine, exhibit a Borel-Anosov Γ and positive φ for which the coarse triangle inequality d_φ(x,z) ≤ d_φ(x,y)+d_φ(y,z)+D fails — in either case Theorem 1.4 and Theorem 4.5 collapse.","supporting_citations":[],"review_version":2}