{"id":"fed85773-6399-41e3-a2ad-dafb7b9d0d8d","arxiv_id":"2608.04966","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Discrete subgroups with critical exponent sufficiently close to the volume growth entropy of the symmetric space are Zariski dense, generalizing Borel's density theorem.","lead":"For any semisimple Lie group with no compact factors, the paper proves that a discrete subgroup whose exponential growth rate is close enough to the maximum must be Zariski dense. The result extends Borel's classical density theorem for lattices and gives explicit numeric thresholds in hyperbolic spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The parabolic-case bound in Theorem 4.4 depends on an unproved black-box (Proposition 4.1) that may not hold uniformly for all proper parabolic Q; the stated epsilon may fail if the almost-L^{2m} exponent degrades with Q.","rationale":"The reader identified Proposition 4.1 as a key black box, and my analysis agrees that this is the most load-bearing concern. The main theorem's existence of epsilon depends entirely on obtaining a strict gap delta_{G/Q} < 1 for every proper parabolic Q, which in turn depends on the almost-L^{2m} property. If this black box is wrong or misquoted, the theorem has no proof. The second load-bearing dependence is the Benoist–Liang identity, also flagged by the reader. I do not see a more fundamental internal inconsistency; the proof structure is otherwise coherent. However, because both of these are unproved citations, the correctness risk is medium. The concrete test I propose would settle whether the black box holds for the key parabolic case, and thus whether the paper's central claim has a valid proof. I therefore maintain the reader's CONDITIONAL verdict rather than upgrading to ACCEPT or downgrading to REJECT.","tokens_in":13060,"tokens_out":920,"duration_ms":12676,"concrete_test":"Verify Proposition 4.1 for the specific case of a proper parabolic subgroup Q of a simple group G, e.g., G = SL(3,R) and Q the minimal parabolic (Borel) subgroup. Concretely, compute or locate in the literature the optimal integrability exponent p_{G/Q} for this case and check whether L^2(G/Q) is almost L^{2m} for some finite m. If no finite m exists (i.e., p_{G/Q} = infinity), then Proposition 4.1 is false and the parabolic bound in Theorem 4.4 collapses. Additionally, check whether Benoist–Liang's Theorem 3.9 (delta_{G/Q} = 1 - 1/p_{G/Q}) has been rigorously proved for all parabolic Q, or if it requires additional hypotheses (e.g., H reductive or discrete) that fail for parabolics.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim, Theorem 4.4, asserts a uniform epsilon(G)>0 such that any discrete Gamma with delta(Gamma) > h_vol(X) - epsilon is Zariski dense. The proof reduces non-Zariski-dense Gamma to either a reductive subgroup (Case 1) or a proper parabolic subgroup (Case 2). The parabolic bound in (4.6) relies on Proposition 4.1, quoted from Einsiedler–Margulis–Venkatesh and Moore, that for every proper closed connected H, L^2(G/H) is almost L^{2m} for some 1 <= m < infinity. This proposition is stated without proof and without a reference to a specific theorem that establishes it for this generality (proper closed connected H, not just parabolic or reductive). If the proposition is false or the exponent m cannot be chosen uniformly over all parabolics (the set Q is finite, so uniformity over Q is automatic, but the result must hold for each Q), then inequality (4.6) could fail and epsilon might not exist. Moreover, the proof of Proposition 4.2 uses Proposition 4.1 for each simple factor, then takes q >= max{2 m_i}; this only transfers an almost-L^{2m} property to tensor products via generalized Holder, but the passage from 'almost L^{2m}' to 'totally L^{2m+}' via Lemma 2.7 requires p >= 2; here m >= 1, so 2m >= 2, ok. A second load-bearing step is the identity delta_{G/H} = 1 - 1/p_{G/H} (Theorem 3.9), which is cited from Benoist–Liang; if this fails for parabolic subgroups, the bound delta_{G/Q} = 1 - 1/p_{G/Q} is unsupported. The paper does not prove this identity, and it is essential for (4.7).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a new 'Borel-type' density theorem for discrete subgroups of connected real semisimple linear algebraic groups without compact factors and with finite center. Theorem 1.2 (restated as Theorem 4.4) asserts that there exists a constant ε = ε(G) > 0 such that any discrete subgroup Γ < G with critical exponent δ(Γ) > h_vol(X) − ε must be Zariski dense in G. The proof separates the non-Zariski-dense subgroups into a reductive case (which reduces to the volume entropy of a proper symmetric subspace) and a parabolic case (which uses the Benoist–Liang identity δ_{G/H} = 1 − 1/p_{G/H} together with a spectral-gap result attributed to Einsiedler–Margulis–Venkatesh and Moore). In real rank one, Theorem 4.10 gives explicit numerical thresholds for SO(n,1), SU(n,1), Sp(n,1), and F_4^{-20}, with an optimality discussion in Remark 4.11. The paper also derives a corollary on the Benoist limit cone (Corollary 4.9).","tokens_in":13445,"tokens_out":24222,"duration_ms":272648,"significance":"If the main theorem is correct, it provides a genuine generalization of Borel's density theorem to infinite-covolume discrete subgroups, using only growth and representation-theoretic input rather than the Corlette–Leuzinger gap property. The construction of a uniform ε(G) from finite maxima over symmetric subspaces and standard parabolics is conceptually clean, and the rank-one quantitative results are potentially useful. The paper is clearly written and the main proof idea is coherent. However, the quantitative theorem contains a false claim in low dimensions, and two technical steps in the proof need to be repaired or better documented. These issues do not necessarily invalidate Theorem 4.4, but they must be addressed before the paper can be accepted.","major_comments":[{"comment":"Theorem 4.10(a) is false for n = 2. For G locally isomorphic to SO(2,1) ≅ PSL(2,R), the cyclic subgroup generated by the unipotent element z ↦ z+1 is discrete and not Zariski dense, yet its critical exponent is 1/2, which is strictly larger than n−2 = 0. The proof's own bound for this case is max(h_vol(X)/2, max_Y h_vol(Y)) = 1/2, so the correct threshold should be δ(Γ) > 1/2, not δ(Γ) > 0. The same error occurs in Theorem 4.10(b) for SU(1,1) (n = 1), where 2n−2 = 0 but the cyclic parabolic subgroup again has critical exponent 1/2. These counterexamples also invalidate the optimality statements in Remark 4.11 for these cases.","section":"Section 4, Theorem 4.10(a) and (b)"},{"comment":"The proof claims that because λ_{G/Q} is strongly L^{2m}, it is 'in particular almost L^{2m}'. This implication is backwards: Definition 2.6(b) defines 'almost L^p' as strongly L^{p+ε} for every ε > 0, and strong L^{2m} alone does not imply strong L^{2m+ε} for arbitrary ε. Since the finiteness of p_{G/Q} in inequality (4.6) relies on this conclusion, the argument is load-bearing. The gap is repairable: the Hölder product estimate works for every q' > q, so λ_{G/Q} is strongly L^{2m'} for all m' > m, and hence almost L^{2m}. As written, however, the proof does not establish the required property.","section":"Section 4, proof of Proposition 4.2"},{"comment":"The parabolic case of Theorem 4.4 depends crucially on Proposition 4.1, which asserts that for every simple non-compact real algebraic group G and every proper closed connected subgroup H, the quasi-regular representation L^2(G/H) is almost L^{2m} for some 1 ≤ m < ∞. The paper states this proposition without proof and only cites [14] and [25] in broad terms. Because the uniform bound in (4.6) — and hence the existence of ε(G) — would fail if this assertion did not hold for all parabolic subgroups Q, the authors should provide a precise theorem number in the cited works or include a direct proof. As it stands, the core result rests on an unverified black box.","section":"Section 4, Proposition 4.1 and proof of Theorem 4.4"}],"minor_comments":[{"comment":"The phrase 'q ≥ max{2m_i}' should presumably be 'q > max{2m_i}', since the 'totally L^{2m_i+}' property guarantees membership in L^q only for strict q > 2m_i.","section":"Section 4, proof of Proposition 4.2"},{"comment":"The claim that there are 'finitely many equivalence classes of isometric Riemannian symmetric subspaces of X' is used to justify the maximum in (4.5), but no reference or argument is supplied; a brief justification or citation would clarify why the maximum is finite.","section":"Section 4, proof of Theorem 4.4"},{"comment":"The abstract and introduction state that the paper 'determines the smallest possible value' of the critical exponent for the real, complex, and quaternionic hyperbolic spaces; this claim requires qualification in view of the corrected thresholds in Theorem 4.10(a) and (b).","section":"Introduction and Remark 4.11"}],"recommendation":"major_revision","confidential_remarks":"The counterexample to Theorem 4.10(a) is elementary and will be quickly noticed by readers; it should be fixed before the paper is considered further. The black-box status of Proposition 4.1 is the most serious concern for the main theorem, and the editor may wish to verify that the cited statement indeed appears in the references cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the central result, a uniform epsilon_G such that delta(Gamma) > h_vol - epsilon forces Zariski density, is a genuine advance. It gives a new proof of Borel for lattices and a corollary on Benoist limit cones. The route through Benoist–Liang temperedness and a spectral gap is a fresh combination, and the paper is mostly clearly written.\n\nWhat is new: Theorem 4.4 and Corollary 4.9. The proof is a clean dichotomy between reductive and parabolic non-density, using delta_{G/H} and p_{G/H}. If the black-box inputs hold, the argument works.\n\nSoft spots. The quantitative rank-one Theorem 4.10(a) is false for n=2: in PSL(2,R), a cyclic parabolic subgroup generated by z -> z+1 is discrete, not Zariski dense, and has critical exponent 1/2, which exceeds n-2 = 0. The proof itself would give the correct bound max(h_vol/2, ...) = 1/2, so the stated threshold (n-2) simply fails to match the argument at low dimension. The optimality remark 4.11 also needs checking at n=2 for (b) and (c); the suggested 'similar arguments' do not obviously produce the claimed threshold when the lower-dimensional subspace has entropy different from the formula. These are fixable but must be fixed.\n\nThe bigger methodological concern is Proposition 4.1. The theorem as stated is loaded: it is attributed to EMV and Moore, but without a precise theorem number or proof. The uniform epsilon in Theorem 4.4 depends on finiteness of Q, which is fine, but it depends on the almost-L^{2m} property holding for each parabolic. If Proposition 4.1 is not in the literature in that generality, the central proof is missing a foundation. The author should either prove it, give a citable statement, or at least state it as an assumption. Also, the identity delta_{G/Q} = 1 - 1/p_{G/Q} from Benoist–Liang is essential and is not proven; that is acceptable as a citation, but its precise scope should be checked.\n\nCitation pattern is fine. The one self-citation [28] is for a standard fact about Quint's growth indicator; that is legitimate.\n\nMy verdict: the main theorem is plausible and worth a serious referee, but the paper needs a revision that corrects the rank-one statements and pins down the black box. If Proposition 4.1 turns out to be false in the stated generality, the parabolic case collapses; the author should be asked to address this head-on. Send it to review, with a caveat that the referee should check the EMV/Moore attribution.","headline":"Main theorem is new and likely true, but the rank-one quantitative section has concrete errors at n=2 and the proof leans on a heavy black box; conditional accept, needs serious revision.","tokens_in":13983,"tokens_out":4901,"would_cite":false,"duration_ms":55865,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","22E46"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for every connected real semisimple linear algebraic group G without compact factors and with finite center, there is a positive epsilon(G) such that any discrete subgroup whose critical exponent exceeds h_vol(X)…","keywords":["Zariski density","critical exponent","semisimple Lie groups","unitary representations","temperedness","lattice density theorem","limit cone","real rank one"],"falsifier":"Exhibit a connected real semisimple linear algebraic group $G$ without compact factors and finite center and a sequence of discrete subgroups $\\Gamma_k<G$, each contained in a proper parabolic subgroup, with critical exponents $\\delta(\\Gamma_k)$ tending to $h_{\\mathrm{vol}}(X)$; the theorem then fails for every $\\epsilon>0$.","tokens_in":12837,"feed_emoji":"📐","tokens_out":10476,"duration_ms":95358,"temperature":0.7,"pith_summary":"This paper establishes a growth-based criterion for Zariski density of discrete subgroups of semisimple Lie groups. For any connected real semisimple linear algebraic group $G$ without compact factors and with finite center, there is a constant $\\epsilon(G)>0$ such that every discrete subgroup $\\Gamma<G$ with critical exponent $\\delta(\\Gamma)>h_{\\mathrm{vol}}(X)-\\epsilon(G)$ is Zariski dense, where $X$ is the symmetric space of $G$ and $h_{\\mathrm{vol}}(X)$ is its volume growth entropy. Since lattices have $\\delta(\\Gamma)=h_{\\mathrm{vol}}(X)$, this recovers the classical density theorem for lattices and extends it to infinite-covolume subgroups whose growth is close to the maximum possible. The proof shows that non-Zariski-dense subgroups must lie in proper reductive or proper parabolic subgroups, and in both cases their critical exponent is bounded by a uniform constant below $h_{\\mathrm{vol}}(X)$. In real rank one, the sharp thresholds are computed: $\\delta(\\Gamma)>n-2$ for $SO(n,1)$, $\\delta(\\Gamma)>2n-2$ for $SU(n,1)$, $\\delta(\\Gamma)>4n-2$ for $Sp(n,1)$, and $\\delta(\\Gamma)>11$ for $F_4^{-20}$.","feed_headline":"Critical exponent near maximum implies Zariski density","feed_subtitle":"A uniform gap below volume entropy extends the lattice density theorem to infinite-covolume subgroups.","key_machinery":"The load-bearing identity is the Benoist-Liang relation $\\theta_{G/H}=\\delta_{G/H}=1-1/p_{G/H}$, which equates the coefficient decay exponent, the relative volume growth exponent, and the reciprocal of the optimal integrability exponent of the quasi-regular representation $L^2(G/H)$. For a discrete subgroup $\\Gamma$ contained in a proper parabolic subgroup $Q$, this identity plus the monotonicity $\\delta_{G/\\Gamma}\\le\\delta_{G/Q}$ gives a bound on the growth indicator $\\psi_\\Gamma$, and the growth indicator theorem guarantees a direction $w\\in\\mathfrak a^+$ with $\\psi_\\Gamma(w)=\\delta(\\Gamma)$. The finiteness of $p_{G/Q}$ is obtained from the almost-$L^{2m}$ property of $L^2(G/Q)$, a consequence of the spectral gap input (Proposition 4.1) combined with the characterization of tempered representations as almost $L^2$.","core_discovery":"The central claim, Theorem 1.2, is that the maximum possible growth rate of a non-Zariski-dense discrete subgroup is uniformly separated from the volume growth entropy of the ambient symmetric space. Specifically, for every connected real semisimple linear algebraic group $G$ with no compact factors and finite center there exists $\\epsilon=\\epsilon(G)>0$ so that if $\\Gamma<G$ is discrete and $\\delta(\\Gamma)>h_{\\mathrm{vol}}(X)-\\epsilon$, then $\\Gamma$ is Zariski dense in $G$. The proof splits the Levi decomposition of a proper algebraic subgroup containing $\\Gamma$: reductive factors preserve a proper totally geodesic symmetric subspace with strictly smaller entropy, while a nontrivial unipotent radical forces $\\Gamma$ into a proper parabolic subgroup $Q$, where the quasi-regular representation $L^2(G/Q)$ is almost $L^{2m}$. The Benoist-Liang identity $\\delta_{G/Q}=1-1/p_{G/Q}$ converts this spectral information into the bound $\\delta(\\Gamma)\\le(1-1/p_{G/Q})h_{\\mathrm{vol}}(X)$, which is strictly below $h_{\\mathrm{vol}}(X)$ because $p_{G/Q}<\\infty$. For real rank one groups the thresholds are sharp for three of the four families, with the optimality for $F_4^{-20}$ left open.","pith_inferences":["An explicit effective value of $\\epsilon(G)$ is not extracted; doing so would require quantitative control of the almost-$L^{2m}$ exponent and of the finitely many integrability exponents $p_{G/Q}$.","The same mechanism might yield Zariski density criteria for other growth notions, such as full directional growth, not just the radial critical exponent.","The open optimality question for $F_4^{-20}$ is essentially whether discrete subgroups of the minimal parabolic subgroup can have critical exponents approaching 11; the paper reports no answer.","One could test the uniformity of the gap numerically in rank one by computing critical exponents of known non-Zariski-dense subgroups and verifying they never exceed the stated thresholds."],"forward_implications":["The classical density theorem for lattices is recovered, since lattices satisfy $\\delta(\\Gamma)=h_{\\mathrm{vol}}(X)$.","If $\\delta(\\Gamma)>h_{\\mathrm{vol}}(X)-\\epsilon(G)$, the limit cone of $\\Gamma$ is a closed convex cone with non-empty interior (Corollary 4.9).","In real rank one, the bounds are sharp for $SO(n,1)$, $SU(n,1)$, and $Sp(n,1)$: a lattice in a proper totally geodesic hyperbolic subspace reaches the stated threshold and is not Zariski dense.","Infinite-covolume subgroups of $Sp(n,1)$ with critical exponent greater than $4n-2$, previously constructed by Corlette, are Zariski dense.","The proof gives a new, representation-theoretic route to Zariski density that does not rely on the Corlette-Leuzinger gap and applies uniformly to groups with and without Kazhdan's property (T)."],"supporting_citations":[{"why":"Supplies the identity $\\delta_{G/H}=1-1/p_{G/H}$ and the relative volume growth exponent used to convert spectral information into critical exponent bounds.","marker":"[9]"},{"why":"Provides the spectral gap result (Lemma 6.5) used to show $L^2(G/Q)$ is almost $L^{2m}$ for proper parabolic subgroups.","marker":"[14]"},{"why":"Gives the exponential decay of matrix coefficients for quasi-regular representations of simple groups, used together with [14] to prove Proposition 4.1.","marker":"[25]"},{"why":"Characterizes tempered representations as almost $L^2$, linking the integrability exponent $p_{G/H}$ to temperedness.","marker":"[12]"},{"why":"Shows almost $L^p$ is equivalent to totally $L^{p+}$, upgrading the almost-$L^{2m}$ property to integrability of all matrix coefficients.","marker":"[29]"},{"why":"Defines the growth indicator and proves there is a direction where it equals the critical exponent, the step that turns the parabolic bound into a critical-exponent bound.","marker":"[26]"},{"why":"Establishes that $h_{\\mathrm{vol}}(X)=2\\|\\rho\\|$ and that lattices have critical exponent $h_{\\mathrm{vol}}(X)$, fixing the normalization used throughout.","marker":"[22]"},{"why":"The density theorem for lattices that the main theorem generalizes and recovers as a special case.","marker":"[10]"}],"fun_headline_variants":["Critical exponent gap forces Zariski density","Near-max entropy implies Zariski density","Sharp threshold for Zariski density","Uniform epsilon below entropy yields density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the imported spectral-gap fact (Proposition 4.1) that for every proper parabolic subgroup $Q$ the quasi-regular representation $L^2(G/Q)$ is almost $L^{2m}$ for some finite $m$; if that failed, the uniform gap $\\epsilon$ would not be guaranteed to exist.","fun_headline_variants_meta":{"raw":{"variants":["Critical exponent gap forces Zariski density","Near-max entropy implies Zariski density","Sharp threshold for Zariski density","Uniform epsilon below entropy yields density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000225,"raw_usage":{"total_tokens":1532,"prompt_tokens":1081,"completion_tokens":451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":399}},"tokens_in":697,"tokens_out":451,"duration_ms":5366,"temperature":1.0,"reasoning_tokens":399,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:52:09.942647+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a connected real semisimple linear algebraic group $G$ without compact factors and finite center and a sequence of discrete subgroups $\\Gamma_k<G$, each contained in a proper parabolic subgroup, with critical exponents $\\delta(\\Gamma_k)$ tending to $h_{\\mathrm{vol}}(X)$; the theorem then fails for every $\\epsilon>0$.","supporting_citations":[{"cited_title":"On the rate of exponential decay of coefficients on homogeneous spaces,","cited_arxiv_id":null,"evidence_quote":"Supplies the identity $\\delta_{G/H}=1-1/p_{G/H}$ and the relative volume growth exponent used to convert spectral information into critical exponent bounds."},{"cited_title":"Effective equidistribution for closed orbits of semisimple groups on homogenous spaces,","cited_arxiv_id":null,"evidence_quote":"Provides the spectral gap result (Lemma 6.5) used to show $L^2(G/Q)$ is almost $L^{2m}$ for proper parabolic subgroups."},{"cited_title":"Exponential decay of correlation coefficients for geodesic flows,","cited_arxiv_id":null,"evidence_quote":"Gives the exponential decay of matrix coefficients for quasi-regular representations of simple groups, used together with [14] to prove Proposition 4.1."},{"cited_title":"AlmostL2 matrix coefficients,","cited_arxiv_id":null,"evidence_quote":"Characterizes tempered representations as almost $L^2$, linking the integrability exponent $p_{G/H}$ to temperedness."},{"cited_title":"ExoticC∗-algebras of geometric groups,","cited_arxiv_id":null,"evidence_quote":"Shows almost $L^p$ is equivalent to totally $L^{p+}$, upgrading the almost-$L^{2m}$ property to integrability of all matrix coefficients."},{"cited_title":"Divergence exponentielle des sous-groupes discrets en rang supérieur,","cited_arxiv_id":null,"evidence_quote":"Defines the growth indicator and proves there is a direction where it equals the critical exponent, the step that turns the parabolic bound into a critical-exponent bound."},{"cited_title":"Kazhdan’sproperty(T),L 2-spectrum, andisoperimetricinequalitiesforlocally symmetric spaces,","cited_arxiv_id":null,"evidence_quote":"Establishes that $h_{\\mathrm{vol}}(X)=2\\|\\rho\\|$ and that lattices have critical exponent $h_{\\mathrm{vol}}(X)$, fixing the normalization used throughout."},{"cited_title":"Density properties for certain subgroups of semi-simple groups without compact components,","cited_arxiv_id":null,"evidence_quote":"The density theorem for lattices that the main theorem generalizes and recovers as a special case."}],"review_version":1}