{"id":"9e454e92-8e14-46c0-a32c-955a7d2770be","arxiv_id":"2608.01382","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For non-lattice discrete subgroups of higher-rank simple Lie groups, the maximal injectivity radius on balls of radius r grows at least c log log log log r.","lead":"This paper proves explicit logarithmic lower bounds for injectivity radius growth in non-compact locally symmetric spaces of higher rank, and gives a new proof of the Stück-Zimmer theorem. It is built on a companion framework of 'almost stationary' measures, which aims to make higher-rank rigidity quantitative.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 2.4(b) yields a proportion constant c_G that depends on Γ; the asserted Γ-independence is not established.","rationale":"The reader's weakest_assumption points to the unpublished companion paper [26] as the structural external dependency; I agree that is a major risk. However, the most load-bearing concern that is checkable from the present manuscript is internal: the proof of Theorem 2.4(b) does not establish the stated Γ-independence of the proportion c_G. The reader's rationale already noted an 'apparent inconsistency in the dependence of the constant c_G,' so we partially agree, but I elevate this to the primary concern because it directly affects the precise central claim and can be settled by tracing constants in the preprint. The imported [26] threatens the whole derivation, but the c_G issue is a concrete gap in the argument as written. Since the gap is repairable by weakening one clause without affecting the main applications, the reader's CONDITIONAL verdict remains appropriate: the paper should be accepted (if at all) only conditional on a correct uniform constant or an explicit weakening of Theorem 2.4(b), and on the availability and correctness of [26].","tokens_in":39599,"tokens_out":7877,"duration_ms":82092,"concrete_test":"Recompute the constant chain in §3 from Lemma 3.6 to (3.20), explicitly tracking the arguments of ε0 and c_G. If, as written, ε0 = ε0(Γ) enters η and hence c_G, then the proof only yields c_G = c(G,Γ). Verify whether any other step supplies a uniform lower bound for ε0(Γ); if none does, weaken Theorem 2.4(b) to “c_G depends on G and Γ” and check that Corollary 1.1 and Section 6 remain valid with this weaker constant. To test whether the stronger uniform claim is actually false, attempt to construct a sequence of non-lattices Γ_n for which the property(T_B) constant of the representation on L^1(G/Γ_n) tends to 0; such a sequence would show the uniform c_G cannot be recovered.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing internal issue is the claimed Γ-independence of c_G in Theorem 2.4(b). In the proof, the author sets η = (ε0/2C)^4, where ε0 is the threshold from Lemma 3.6. But Lemma 3.6 explicitly states ε0 = ε0(Γ), and its proof obtains ε0 from property(T_B) applied to L^1(G/Γ, m_{G/Γ}); the resulting Kazhdan-type constant depends on the quasi-invariant measure, hence on Γ. Equation (3.20) then defines c_G := (1/32)(ε0/2)^4/(2C), so c_G inherits the Γ-dependence. Nothing in Sections 3–4 supplies a uniform positive lower bound for ε0(Γ) over all discrete infinite-covolume subgroups Γ. Thus the statement “where c_G > 0 depends only on G” is not proved; the argument establishes at most a constant c(G,Γ). This is load-bearing because part (b) is the “definite proportion” claim. The flaw does not destroy Theorem A or the Stück–Zimmer application, which only need a positive proportion for each fixed Γ, but it means the theorem as stated is stronger than the proof supports.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops quantitative versions of higher-rank rigidity results for discrete subgroups of noncompact simple Lie groups. Its main theorem (Theorem 2.4) asserts that if a discrete subgroup Γ has infinite covolume, then in every sufficiently large ball of X/Γ there is a point with injectivity radius at least c log^(4) r, and a positive proportion of the points visited by a random walk of length r have this property. Corollary 1.1 derives a lattice-rigidity statement: if maximal injectivity radius on balls grows slower than log^(4) r, then Γ is a lattice. The paper also proves a new characterization of lattices via the existence of an approximately invariant measure (Theorem B / Lemma 3.6) and gives a one-page proof of the Stück–Zimmer theorem for ergodic actions of higher-rank simple groups with property (T), replacing the traditional Nevo–Zimmer/intermediate-factor machinery with an effective almost-stationary structure theorem. The appendices contain substantial analytic inputs: heat-kernel log-Lipschitz estimates, random-walk concentration, and a convolution-power Margulis-type inequality, all proved in the text.","tokens_in":39979,"tokens_out":6688,"duration_ms":64056,"significance":"If the main theorem holds as stated, it is a substantial quantitative advance over the qualitative result of Fraczyk–Gelander, and the short Stück–Zimmer proof is an elegant demonstration of the power of the almost-stationary method. The paper is careful in many places: the random-walk, heat-kernel, and discreteness-radius appendices are detailed and self-contained, and the geometric part of the argument is written with a clear structural strategy. The main obstruction to acceptance is the heavy reliance on the companion manuscript [26], which is cited as 'Manuscript in preparation' and is not available. A second, genuinely internal issue is that the constant c_G in Theorem 2.4(b) is not proved to depend only on G; the proof gives c_G = c_G(Γ). Both issues are load-bearing for the stated main theorem, but both are in principle repairable. The central geometric ideas appear sound for the weaker formulation with constants depending on Γ.","major_comments":[{"comment":"The main engine of the paper is Theorem 3.5, the 'almost Nevo–Zimmer' dichotomy, which is stated as a black box from [26, Theorem 1.10], an unpublished companion manuscript. It is used directly in the proof of Theorem 2.4 to split into the almost-invariant case and the almost-projective-factor case, and it is also needed for the Stück–Zimmer proof through Theorem 2.4. In addition, Lemma 5.16 begins with 'The fact that ν̃_m can be written in this form follows from the almost Furstenberg decomposition [26]', so the almost-Furstenberg decomposition is another imported, unproved input. As submitted, the central theorem cannot be independently checked. I am not questioning the validity of the companion results, but for a published proof the dependency must be verifiable. The authors should either include the relevant statements and proofs of the companion results, post the companion manuscrip","section":"§3, Theorem 3.5; §5, Lemma 5.16"},{"comment":"Theorem 2.4(b) asserts that the proportion c_G depends only on G. In the proof, however, the constant η is fixed as η=(ε0/(2C))^4, where ε0=ε0(Γ) is the threshold from Lemma 3.6. Lemma 3.6 states ε0=ε0(Γ), and its proof obtains ε0 from property (T_B) applied to L^1(G/Γ,m_{G/Γ}); the resulting Kazhdan-type constant depends on the quasi-invariant measure, hence on Γ. Equation (3.20) then sets c_G := (1/32)(ε0/2)^4/(2C), so c_G inherits this Γ-dependence. Nothing in Sections 3–4 gives a uniform positive lower bound for ε0(Γ) over all discrete infinite-covolume subgroups Γ. Thus the assertion 'where c_G > 0 depends only on G' is not proved. The argument establishes at most a constant c(G,Γ). This is load-bearing for the statement of part (b), but not for the main Theorem A or the Stück–Zimmer application, which only require a positive proportion for each fixed Γ. I recommend either proving t","section":"Theorem 2.4(b) and proof, Eq. (3.20)"},{"comment":"The proof of Lemma 5.1 goes through Lemma 5.13, whose hypothesis includes a norm-δ, P-invariant measure λ. The construction of λθ in Lemma 5.16 gives measures that are m^{-1/2}-P-almost invariant, not exactly P-invariant. In the proof of Lemma 5.1, this is accounted for by taking δ=m^{-1/2}; however, Lemma 5.13 is stated and proved under exact P-invariance, and the perturbation argument that would justify applying it to an almost invariant measure is only sketched. I am not claiming the gap is fatal, but the formal status of the 'almost P-invariance to exact invariance' passage should be stated clearly as an approximation step, with the resulting error terms tracked through Claims 5.10 and 5.11.","section":"§5, Lemma 5.13 and proof of Lemma 5.1"}],"minor_comments":[{"comment":"The abstract's criterion for lattices uses W_1^b(gν,ν) ≤ ε0, whereas Theorem B in §1.3 and Lemma 3.6 require a stronger condition involving the observable f_{ν,g} = dν/d(gν), plus Lipschitz and bounded-density assumptions. The abstract should match the precise statement, or explicitly call itself a simplified version.","section":"Abstract"},{"comment":"The sentence 'Since δ_{e} is a G-invariant measure and since ν was assumed ergodic, we obtain that ν=δ_{e} as desired' appears twice verbatim. Please remove the duplication.","section":"§6, proof of Theorem 1.2"},{"comment":"There are small typos: 'Cézaro' should be 'Cesàro', 'at last on' should be 'for at least one of', and 'injection radius' should be 'injectivity radius' where it appears.","section":"Throughout"},{"comment":"The Lambert W-function is introduced and not used in an essential way in the main text; the explicit formula following Eq. (3.16) is hard to read. It would help to state the bound in asymptotic form directly, which is already done in Eq. (3.17).","section":"§2, Definition 2.3 and §3.5"},{"comment":"Reference [50] appears to cite a Math StackExchange user name ('stephantu') rather than a standard citable source. If the fact is standard, replacing this reference by a textbook or by a proof in the appendix would be more appropriate.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The reliance on [26] is structural and substantial. If the companion manuscript is to be submitted as a separate paper, I would recommend that the editor require either that it be posted and publicly available before final acceptance, or that the key theorems be reproduced in the appendix of the present paper. The c_G issue is more internal and can be fixed by a modest weakening or a uniform argument. With those changes the paper's central geometric contribution would be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a serious piece of work with real new results, but the main theorem can't be verified from this preprint alone, and Theorem 2.4(b) is stated more strongly than the proof supports. Conditional on the companion [26] holding up, I think the arguments are largely sound.\n\nWhat's new: the quantitative log^(4) growth of injectivity radius along balls, the almost-invariance criterion for lattices (Theorem B), and a short new proof of Stück-Zimmer. The appended GLM-type inequality (Theorem D.2) for bi-K-invariant measures with exponential moment is also a genuine standalone contribution, and the appendices on log-Lipschitz density and heat-kernel concentration look careful. I also appreciate that the authors are upfront about where the four logarithms come from and that log^(4) is probably not optimal.\n\nThe soft spots are real. First, the engine of the paper — Theorem 3.5, the almost Nevo-Zimmer dichotomy — is quoted from the authors' own unpublished companion [26]. That makes the main results impossible to check from this preprint alone. This is not a minor omission; it's the load-bearing part. Second, the stress-test note is right: the proof of Theorem 2.4(b) sets η=(ε0/2C)^4 with ε0 from Lemma 3.6, and ε0 is ε0(Γ). Then c_G inherits that dependence. So the claim that c_G depends only on G is not established. Interestingly, the abstract says c=c(G,Γ), so the paper is internally inconsistent about this. The fix is straightforward — state the theorem with c(G,Γ) — and the rest of the paper (Theorem A, Stück-Zimmer) only needs a positive proportion for each fixed Γ, so the core results survive.\n\nOne more thing: Section 6's proof of Stück-Zimmer uses Theorem 2.4, but that's fine since it already proved the discrete case for fixed Γ. My main reservation is the dependency on [26], not the internal math. If [26] doesn't exist in the stated form, the main theorems don't go through.\n\nBottom line: this deserves a serious referee, and the journal should ask the authors to post [26] or include the needed statements as an appendix. The constant issue should be fixed before publication.\n\nRecommendation: send to peer review, with a note to the authors about the c_G dependence, and require the companion to be available.","headline":"Real results, but the central tool is a black box from an unpublished companion, and the advertised Γ-independence of c_G in Theorem 2.4(b) is not established by the proof.","tokens_in":40380,"tokens_out":2773,"would_cite":false,"duration_ms":26423,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","37A17","53C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a non-lattice discrete subgroup Γ of a higher-rank simple Lie group forces the injectivity radius of its locally symmetric space X/Γ to grow at least like c log-log-log-log r on growing balls, and characterizes lattic","keywords":["injectivity radius","higher-rank Lie groups","almost stationary measures","random walks on homogeneous spaces","Stück–Zimmer theorem","lattice rigidity","property (T)","locally symmetric spaces"],"falsifier":"Exhibit a discrete non-lattice subgroup $\\Gamma$ of $SL(3,\\mathbb{R})$ and a sequence $r_n\\to\\infty$ for which the maximal injectivity radius on the ball of radius $r_n$ grows slower than $c\\log^{(4)} r_n$ for every $c>0$; Corollary 1.1 asserts no such $\\Gamma$ exists. A computable candidate would be a finitely generated free subgroup whose quotient's injectivity radius can be numerically traced along random-walk orbits, checking whether the iterated-logarithm growth is violated.","tokens_in":39542,"feed_emoji":"📏","tokens_out":7892,"duration_ms":70183,"temperature":0.7,"pith_summary":"The paper proves a quantitative rigidity theorem for discrete subgroups of higher-rank simple Lie groups. It shows that a non-lattice subgroup Γ forces the injectivity radius of the associated locally symmetric space to grow at least like $c\\log^{(4)} r$ along balls of radius $r$, with a fixed positive proportion of random-walk steps landing at such points. Equivalently, if the maximal injectivity radius on balls grows slower than four iterated logarithms, Γ must be a lattice — a sharp dichotomy with no intermediate growth rate. The proof replaces the classical passage to invariant limit measures with almost stationary Cesàro averages, yielding an explicit gap phenomenon: an almost invariant probability measure on $G/\\Gamma$ exists only when Γ is a lattice. As a by-product, the method gives a one-page proof of the Stück–Zimmer theorem.","feed_headline":"Non-lattices force four-log injectivity radius growth","feed_subtitle":"If the largest embedded balls don't grow like four stacked logs, the subgroup must be a lattice.","key_machinery":"The load-bearing mechanism is the 'almost Nevo–Zimmer dichotomy' (Theorem 3.5), imported from the authors' companion manuscript [26]: an $\\epsilon$-almost stationary probability measure on a $G$-space is either almost invariant (up to an error polynomially small in $|\\log\\epsilon|^{-1}$) or has a definite proportion of its mass concentrated on points whose stabilizers are approximately contained in a proper parabolic subgroup. The paper feeds into this dichotomy the Cesàro averages $\\nu_n = \\frac1n\\sum_{i=1}^n \\mu^{*i}*\\delta_x$, which are automatically $2/n$-almost stationary with no limit passage. Case (a) is ruled out by Lemma 3.6 (Theorem B), which uses property (T) in its $L^1$ form to","core_discovery":"The central claim is Theorem 2.4: for every connected simple Lie group $G$ with real rank at least 2 and every discrete infinite-covolume subgroup $\\Gamma$, there are constants $c>0$ and $r_0$ such that for every $r\\ge r_0$ the ball of radius $r$ in $X/\\Gamma$ contains a point of injectivity radius at least $c\\log^{(4)} r$, and a definite proportion $c_G$ of the Cesàro random-walk steps visit such points. Corollary 1.1 restates this as an exact dichotomy: either $\\Gamma$ is a lattice and the injectivity radius is bounded, or the maximal injectivity radius on balls grows at least like a positive multiple of $\\log^{(4)} r$. The paper also establishes a lattice criterion (Theorem B): if a proba","pith_inferences":["The four iterated logarithms are almost certainly not optimal; the explicit accounting for each logarithm suggests a roadmap for shaving them off, and a growth of order $\\log r$ might be the true threshold.","The same almost-stationary machinery should apply to other geometric or spectral quantities previously accessible only through limit measures, such as effective normality of subgroups or quantitative Betti-number convergence without congruence structure.","Because the lattice criterion only needs weak almost invariance of a single measure, its contrapositive provides a potential numerical certificate: an explicit measure on $G/\\Gamma$ with $W^f_1$-distance below $\\epsilon_0$ would prove $\\Gamma$ is not a lattice.","If the companion manuscript's dichotomy is extended to cover free actions or a wider class of step measures, the Stück–Zimmer theorem and the injectivity-radius bounds would carry over to settings not covered by the current statement."],"forward_implications":["Corollary 1.1 is a dichotomy with no intermediate growth: a discrete subgroup $\\Gamma$ is a lattice if and only if the maximal injectivity radius on balls of radius $r$ is $o(\\log^{(4)} r)$.","The lattice criterion (Theorem B) gives a new gap phenomenon: on $G/\\Gamma$ for non-lattice $\\Gamma$, no probability measure can be almost invariant below a threshold $\\epsilon_0(G,\\Gamma)$; approximate invariance forces exact invariance.","Theorem 2.4 quantitatively strengthens Fraczyk–Gelander's resolution of Margulis' conjecture, providing explicit iterated-logarithmic growth of unbounded injectivity radius.","The Stück–Zimmer theorem follows by a one-page argument from the almost stationary dichotomy, the reduction to invariant random subgroups, and the lattice criterion.","The paper announces (in work in preparation) that the constant $c_G$ can be taken arbitrarily close to 1, yielding the first explicit threshold in Benjamini–Schramm convergence for arbitrary sequences of lattices."],"supporting_citations":[{"why":"Supplies the almost Nevo–Zimmer dichotomy (Theorem 3.5) and the almost Furstenberg decomposition that the entire argument runs on.","marker":"[26]"},{"why":"Provides the qualitative theorem being made quantitative and the confined-subgroup strategy with parabolic and Levi steps that Lemmas 3.7 and 5.13 adapt.","marker":"[23]"},{"why":"Property (T) in L^1 form, used in Lemma 3.6 to upgrade weak almost invariance to existence of an invariant measure, forcing the lattice conclusion.","marker":"[10]"},{"why":"Gelander–Levit–Margulis effective discreteness-radius inequality, used in Lemmas 3.8 and 5.14 and proven in convolution form in Appendix D.","marker":"[29]"},{"why":"The Stück–Zimmer theorem whose original proof is replaced by the short argument in Section 6.","marker":"[51]"},{"why":"Reduction of the Stück–Zimmer theorem to classification of ergodic invariant random subgroups via the Abért–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet framework.","marker":"[2]"},{"why":"Benoist–Quint law of large numbers for the Cartan projection, used to prove concentration of the heat kernel around its drift and the convolution-power GLM inequality.","marker":"[16]"},{"why":"Kakutani's random ergodic theorem, used to obtain generic points $x$ for which Cesàro averages converge to the ergodic measure in the Stück–Zimmer proof.","marker":"[32]"}],"fun_headline_variants":["Non-lattices force four-log radius growth","Four-log injectivity growth: non-lattices only","Non-lattices: radius grows like four logs","Non-lattice subgroups guarantee four-log injectivity growth","Non-lattice signature: four-log radius growth"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The proof rests on the almost Nevo–Zimmer dichotomy for $\\epsilon$-almost stationary measures (Theorem 3.5 of this paper), which is proved in the companion manuscript [26]; if that dichotomy fails in the stated form—for instance, if an almost stationary measure could sit in a third case between almost invariance and almost projective factors—the main theorems of this paper do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Non-lattices force four-log radius growth","Four-log injectivity growth: non-lattices only","Non-lattices: radius grows like four logs","Non-lattice subgroups guarantee four-log injectivity growth","Non-lattice signature: four-log radius growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001333,"raw_usage":{"total_tokens":5361,"prompt_tokens":951,"completion_tokens":4410,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":4335}},"tokens_in":695,"tokens_out":4410,"duration_ms":32991,"temperature":1.0,"reasoning_tokens":4335,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:15:47.087515+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a discrete non-lattice subgroup $\\Gamma$ of $SL(3,\\mathbb{R})$ and a sequence $r_n\\to\\infty$ for which the maximal injectivity radius on the ball of radius $r_n$ grows slower than $c\\log^{(4)} r_n$ for every $c>0$; Corollary 1.1 asserts no such $\\Gamma$ exists. A computable candidate would be a finitely generated free subgroup whose quotient's injectivity radius can be numerically traced along random-walk orbits, checking whether the iterated-logarithm growth is violated.","supporting_citations":[{"cited_title":"The almost structure of measures","cited_arxiv_id":null,"evidence_quote":"Supplies the almost Nevo–Zimmer dichotomy (Theorem 3.5) and the almost Furstenberg decomposition that the entire argument runs on."},{"cited_title":"Infinite volume and infinite injectivity radius.Ann","cited_arxiv_id":null,"evidence_quote":"Provides the qualitative theorem being made quantitative and the confined-subgroup strategy with parabolic and Levi steps that Lemmas 3.7 and 5.13 adapt."},{"cited_title":"Property (T) and rigidity for actions on Banach spaces.Acta Math., 198(1):57–105, 2007","cited_arxiv_id":null,"evidence_quote":"Property (T) in L^1 form, used in Lemma 3.6 to upgrade weak almost invariance to existence of an invariant measure, forcing the lattice conclusion."},{"cited_title":"Effective discreteness radius of stabilizers for stationary actions.Michigan Math","cited_arxiv_id":null,"evidence_quote":"Gelander–Levit–Margulis effective discreteness-radius inequality, used in Lemmas 3.8 and 5.14 and proven in convolution form in Appendix D."},{"cited_title":"Stabilizers for ergodic actions of higher rank semisimple groups.Annals of Mathematics, 139 (3)(2):723–747, 1994","cited_arxiv_id":null,"evidence_quote":"The Stück–Zimmer theorem whose original proof is replaced by the short argument in Section 6."},{"cited_title":"On the growth ofL 2-invariants for sequences of lattices in Lie groups.Ann","cited_arxiv_id":null,"evidence_quote":"Reduction of the Stück–Zimmer theorem to classification of ergodic invariant random subgroups via the Abért–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet framework."},{"cited_title":"Springer-Verlag, Berlin, 2016","cited_arxiv_id":null,"evidence_quote":"Benoist–Quint law of large numbers for the Cartan projection, used to prove concentration of the heat kernel around its drift and the convolution-power GLM inequality."},{"cited_title":"RandomergodictheoremsandMarkoffprocesseswithastable distribution","cited_arxiv_id":null,"evidence_quote":"Kakutani's random ergodic theorem, used to obtain generic points $x$ for which Cesàro averages converge to the ergodic measure in the Stück–Zimmer proof."}],"review_version":1}