{"id":"17a29bee-653b-4ee5-a897-39cf8e8f5c30","arxiv_id":"2507.22266","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For arithmetic lattices in products of PGL2(R) and PGL2(C), Zariski-dense subgroups have normalized height bounded below by a constant times log(covolume) divided by the square of the field degree.","lead":"This paper proves a stronger version of a known theorem about the 'height' of finite sets of matrices that generate large groups: for arithmetic lattices built from PGL2, the height must grow at least like a constant times the logarithm of the lattice's covolume. This yields a sharper Margulis lemma and short new proofs of known finiteness and convergence results in hyperbolic geometry and group theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is unproved for lattices whose field of definition has no ramified real places (e.g., Bianchi groups in PGL2(C)): the bounded-degree case is explicitly deferred in §4.3 and §5 but never supplied, and the ε-close conjugates α5,...,αd used there do not exist when [ℓ:Q]=4.","rationale":"The reader's CONDITIONAL verdict is appropriate: the discriminant step is indeed load-bearing, and the weakest point is the unjustified transition from small Galois conjugates to the bounds on log|α| and log Δℓ. My stress-test makes this more precise: in the bounded-degree case, and specifically for imaginary quadratic fields of definition in G=PGL2(C), the small conjugates used in §4.3 and §5 do not exist, and the promised bounded-degree argument is never given. This is a genuine gap in the proof as written, not merely a terse estimate, and it affects a central family of examples (Bianchi groups). The gap may be fillable by a separate fixed-degree argument, but the paper does not provide one, so the main theorem is not fully verified. I do not see an internal inconsistency or a counterexample, only an incomplete proof, so the correct verdict remains CONDITIONAL rather than ACCEPT, REJECT, or UNVERDICTED.","tokens_in":14317,"tokens_out":26770,"duration_ms":321142,"concrete_test":"Analytic test: take k=Q(√−D), G=PGL2(C), so [ℓ:Q]=4 and Sr=∅ in §4.3. Attempt to re-derive Theorem 1.1 for this case using only the estimates available in §4.3–§4.5 without the assumption that α5,...,αd are ε-close to 1. In particular, check whether inequality (7), log|α| ≥ (1/4d)log Δℓ, follows from Δℓ≥1 and the product formula when d=4, and whether the volume bound can be obtained from the direct norm estimate |Nk/Q(α+α−1−2)| ≤ c|α|4 instead of the discriminant route. If either step cannot be completed, the paper contains no proof for Bianchi groups.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 depends on the discriminant estimate in §4.3 (and its analogue in §5) that the almost-law word W has d−4 Galois conjugates α5,...,αd with |αi−1|≤ε. These conjugates come from embeddings of k at which the quaternion algebra is ramified and Γ is unbounded? In fact, they come from real places of k where A is ramified, so σ(Γ) lies in a compact group and the almost law applies. For an imaginary quadratic field k (the standard field of definition of Bianchi groups in PGL2(C)), there are no such real places: d=[ℓ:Q]=4 and the set α5,...,αd is empty. The proof in §4.3 explicitly assumes d≥D0 and says the bounded-degree case will be dealt with 'later on,' but no such argument appears anywhere in the paper; §5 repeats the same assumption. For d=4 the key estimates (3) and (5) become vacuous, so the assertion 'we can assume log|α|>10' and the bound log|α| ≥ (1/4d)log Δℓ are unsupported. Since Bianchi subgroups are explicitly discussed in Remark 1.2 and are a central family of arithmetic lattices in PGL2(C), Theorem 1.1 is not established for them as written. The same missing bounded-degree case also affects subgroups of PGL2(R) over k=Q, where d=2 and there are no small conjugates.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a strong height gap theorem for arithmetic lattices in G = PGL2(R)^a × PGL2(C)^b, a+b ≥ 1. The main result, Theorem 1.1, asserts that for any arithmetic lattice Γ defined over a field k and any finite F ⊂ Γ generating a Zariski dense subgroup, the normalized height satisfies ĥ(F) > c_G max( log(covol(Γ1)) / [k:Q]^2, 1 ), where Γ1 is a congruence subgroup containing Γ. The proof combines the almost-law method of Chen–Hurtado–Lee, an algebraic genericity criterion, a discriminant estimate for a generic element W ∈ Γ(2), and Borel's volume formula. The paper also derives a strong arithmetic Margulis lemma (Corollary 1.6) and applies it to finiteness of arithmetic maximal reflection groups and to Benjamini–Schramm convergence of arithmetic hyperbolic manifolds.","tokens_in":14659,"tokens_out":4618,"duration_ms":53213,"significance":"If the main theorem is correct, it is a significant strengthening of the height gap theorem in the PGL2 setting, with useful consequences such as a covolume-dependent arithmetic Margulis lemma. The proof strategy is fresh and mostly self-contained, and the applications in Section 6 would give short new proofs of known results. However, the proof as written contains a substantial gap: the bounded-degree case, which includes Bianchi groups in PGL2(C) and arithmetic Fuchsian groups over Q, is explicitly deferred in Section 4.3 and again in Section 5, but no argument for it is supplied. Since these families are central to the paper's claims and applications, the theorem is not established as stated.","major_comments":[{"comment":"The proof of the discriminant estimate assumes d ≥ D0 and states that the bounded-degree case 'is much simpler and we will deal with it later on,' but no such treatment appears anywhere in the paper. For Bianchi groups in PGL2(C), the field k is imaginary quadratic, so ℓ = Q(α) has degree d = 4 and the set α5,...,αd is empty. In that case the sum Sr is empty, inequality (3) is vacuous, inequality (5) reduces to S1 ≥ 0, and the argument gives neither log|α| ≥ c(d−4) nor the covolume-dependent lower bound. Since Bianchi subgroups are explicitly discussed in Remark 1.2 as a motivating family, Theorem 1.1 is not proved for them as written.","section":"Section 4.3, Eqs. (3)–(5)"},{"comment":"For G = PGL2(R), the proof is left as an exercise for the reader. This is not an acceptable way to prove one of the cases of the main theorem, especially because the smallest-degree case k = Q, d = 2 has no conjugates α5,...,αd at all and hence requires a separate argument of exactly the kind that is missing in Section 4.3. The omission matters for the applications: the proof of Theorem 6.1 and the reflection-group application in Section 6.1 rely on the strong arithmetic Margulis lemma for PGL2(R), which in turn rests on Theorem 1.1 for that group.","section":"Section 4.5"},{"comment":"The product case repeats the same bounded-degree gap. The text assumes d ≥ D0, sets d0 = 2a+4b+1, and asserts that the αi with i ≥ d0 are ε-close to 1; then it says the bounded-degree case 'will be seen later on,' but no such argument follows. Moreover, the assertion 'we can assume log|α| ≥ c(ε) > 10' is stated as a consequence of the product formula without a derivation. For fields with few or no real places, the set of ε-close conjugates may be empty, so the estimates (12)–(14) do not hold and the subsequent conclusion log|α| ≥ c7 d and log|α| ≥ (c8/d) log Δℓ is unsupported. Since Theorem 1.1 covers the full family of products, this gap affects the main claim in the general case.","section":"Section 5, Eqs. (12)–(14)"}],"minor_comments":[{"comment":"There are several typos and formatting issues: 'simi-simple' in Corollary 1.6 should be 'semi-simple'; 'Fuchisan groups' in Remark 1.3 should be 'Fuchsian groups'; and the displayed formula in Section 4.5 containing '2log2 |Nk/Q(β)||Nk/Q(β)|' is malformed and should be rewritten.","section":"Throughout"},{"comment":"The sentence 'Because w is an almost law we have that α5,...,αd are very close to 1' is too terse: the mechanism by which the almost law on a compact group forces conjugates at ramified real places to be close to 1 is not explained, and the reader has to infer the intended argument from [8].","section":"Section 4.3"},{"comment":"The proof invokes 'a variant of [7, Corollary 1.7] derived from Theorem 1.1' without stating the variant or indicating how it follows from Theorem 1.1; this step should be made explicit.","section":"Proof of Corollary 1.6"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the missing bounded-degree case, which appears to be essential for Bianchi groups and for PGL2(R)-lattices over Q. This is not a matter of presentation; it is a load-bearing gap in the proof of Theorem 1.1. The paper's applications depend on the same missing cases. If the authors can supply a complete argument for the bounded-degree case and for PGL2(R), the paper would be a strong contribution. I found no indication of circularity: the proof uses known tools and does not assume the height gap theorem being strengthened."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a serious attempt to upgrade Breuillard's height gap theorem to a covolume-dependent bound for arithmetic lattices in G = PGL2(R)^a × PGL2(C)^b. The main idea—run the Chen–Hurtado–Lee almost-laws argument, use a generic word to extract an eigenvalue whose height is controlled by the discriminant, then convert discriminant to covolume via Borel's volume formula—is genuinely new, and the PGL2(C) large-degree case is worked out carefully. The applications (strong arithmetic Margulis lemma, new proofs of finiteness of arithmetic reflection groups and Benjamini–Schramm convergence) are real and the methods are appropriate.\n\nThe soft spots are also real. The bounded-degree case is explicitly deferred in §4.3 (\"we will deal with it later on\") and in §5, but it never appears. For Bianchi groups over imaginary quadratic fields, d=[ℓ:Q]=4, the set α5,...,αd is empty, the negative sum Sr is zero, and the inference \"we can assume log|α|>10\" has no support. The same gap hits PGL2(R) over k=Q, where d=2; and the PGL2(R) argument is left as \"an exercise\" anyway. As written, Theorem 1.1 is not established for these standard families. The stress-test note is right about this.\n\nThere are also a few spots where the paper hand-waves: the assertion log|α| ≥ c(ε) > 10 from the product formula is not fully derived; the index bounds [Γ0:Γ1] ≤ c|α|^4 and [Γ1:Γ(I)] ≤ c|α|^6 are asserted after a short \"we conclude\"; and Section 5's \"more crude estimates\" are terse. These are probably fixable, but a referee will need the details.\n\nNone of this kills the central idea. A likely repair for the bounded-degree case is to combine Breuillard's constant gap with the discriminant bound to get the max(log covol, 1) form. So I would not desk-reject; the paper deserves referee time. My guess is the main theorem is true, but the proof as currently written is incomplete. The audience is arithmetic groups and hyperbolic manifolds people; they will want this result. Give it a serious referee, but insist the authors fill the bounded-degree case and write out PGL2(R) before publication.\n\nRecommendation: send to peer review, conditional on completion of the missing cases.","headline":"Covolume-dependent height gap with a good idea and a real gap: bounded-degree cases (Bianchi groups, k=Q) are deferred and never supplied, so Theorem 1.1 is not established as written, but the fix looks simple and the paper warrants refereeing.","tokens_in":15205,"tokens_out":5859,"would_cite":true,"duration_ms":64919,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E40","11F06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For arithmetic lattices in $G = PGL_2(\\mathbb{R})^a \\times PGL_2(\\mathbb{C})^b$, any Zariski-dense finite set $F \\subset \\Gamma$ has normalized height at least $c_G \\max(\\log \\operatorname{covol}(\\Gamma_1)/[k:\\mathbb{Q}]^2, 1)$, with…","keywords":["height gap theorem","arithmetic lattices","covolume","PGL2","Zariski dense subgroups","almost laws","discriminant estimates","arithmetic Margulis lemma"],"falsifier":"Evaluate the double-commutator almost-law word on small Zariski-dense generating sets of $PSL_2(\\mathbb{Z}[\\sqrt{-D}])$ with $D \\to \\infty$ and compute $\\log|\\alpha|$, the degree $d$, and $\\log\\Delta_\\ell$; if for some word length the non-large Galois conjugates fail to cluster within $\\epsilon$ of $1$, or $\\log|\\alpha|$ falls below $c_1(d-4)$ while $\\langle F\\rangle$ remains Zariski dense over $\\mathbb{R}$, then the discriminant chain that links the height to $\\log\\operatorname{covol}(\\Gamma_1)$ collapses. A weaker but immediate check is to test the asserted implication $S_r \\le -c(d-4)(d-5)/2 \\Rightarrow \\log|\\alpha| \\ge c_1(d-4)$ directly on numerical discriminants, since the intermediate derivation of $\\log|\\alpha| > 10$ is not printed.","tokens_in":14094,"feed_emoji":"📐","tokens_out":17905,"duration_ms":172225,"temperature":0.7,"pith_summary":"The paper proves a quantitative strengthening of the height gap theorem for the groups $G = PGL_2(\\mathbb{R})^a \\times PGL_2(\\mathbb{C})^b$ with $a+b \\ge 1$. The classical theorem guarantees that a finite set of matrices generating a non-virtually-solvable subgroup has normalized height bounded below by a constant depending only on the ambient group. This paper shows that when the finite set lies in an arithmetic lattice $\\Gamma$, the bound can be made to grow with the lattice: a Zariski-dense set $F$ must satisfy $\\widehat{h}(F) > c_G \\max(\\log(\\operatorname{covol}(\\Gamma_1))/[k:\\mathbb{Q}]^2, 1)$, where $\\Gamma_1$ is a congruence (for instance, maximal) subgroup containing $\\Gamma$ and $k$ is the field of definition. The interest is that a purely qualitative gap becomes a volume-dependent one, which yields a strong arithmetic Margulis lemma and short proofs of two results previously reached through harder spectral or representation-theoretic inputs.","feed_headline":"Height gap for arithmetic lattices grows with covolume","feed_subtitle":"Zariski-dense sets in PGL2 arithmetic lattices get height at least c log(covolume), sharpening the classical gap.","key_machinery":"Three ingredients carry the argument. (1) Generic elements: $W \\in \\Gamma^{(2)}$ whose eigenvalue $\\alpha$ satisfies $\\mathbb{Q}(\\alpha + 1/\\alpha) = k$ and whose components are all hyperbolic or loxodromic; Lemma 3.2 shows the non-generic elements lie in a proper real subvariety. (2) Almost laws: an $\\epsilon$-almost law $w_0$ on the compact group $(O_3)^a \\times (O_4)^b$, composed into the word $w(A,B) = w_0([A^2,[B^2,A^2]], A^{-1}[A^2,[B^2,A^2]]A)$, whose double commutator guarantees the values lie in $\\Gamma^{(2)}$; together with uniform exponential growth this locates a generic value $W$ inside a bounded power $F^{n_w}$ of $F$. (3) The discriminant estimate: with $d = [\\ell:\\mathbb{Q}]$ and the conjugates ordered so that $\\alpha_5,\\dots,\\alpha_d$ are $\\epsilon$-close to $1$, the identity $\\log\\Delta_\\ell = S_1 + S_2 + S_r$ separates the terms touching the large conjugates, the small conjugates, and the bulk $S_r \\le -c(d-4)(d-5)/2$; the product formula then forces $\\log|\\alpha| \\ge c_1(d-4)$ and $\\log|\\alpha| \\ge (1/(4d))\\log\\Delta_\\ell$. The volume formula of [6] and the congruence and index estimates from the tree action (generalized index $[\\Gamma_0:\\Gamma_1] \\le c_1|\\alpha|^4$, and $[\\Gamma_1:\\Gamma(I)] \\le c_5|\\alpha|^6$ for a congruence subgroup $\\Gamma(I)$) finish the chain.","core_discovery":"On the paper's own terms, Theorem 1.1 asserts: for $G = PGL_2(\\mathbb{R})^a \\times PGL_2(\\mathbb{C})^b$ with $a+b \\ge 1$ there is a constant $c_G > 0$ such that for any arithmetic lattice $\\Gamma \\subset G$ defined over a number field $k$ and any finite $F \\subset \\Gamma$, either $\\langle F\\rangle$ is not Zariski dense over $\\mathbb{R}$, or $\\widehat{h}(F) > c_G \\max(\\log(\\operatorname{covol}(\\Gamma_1))/[k:\\mathbb{Q}]^2, 1)$, where $\\Gamma_1$ is a congruence subgroup, for instance a maximal lattice, containing $\\Gamma$. The proof produces a generic element $W$ from an almost-law word on a compact group: uniform exponential growth guarantees that some value of the word on a bounded power of $F$ is generic, meaning its eigenvalue $\\alpha$ generates a quadratic extension $\\ell = \\mathbb{Q}(\\alpha)$ of $k$ and behaves hyperbolically or loxodromically in every factor. The discriminant $\\Delta_\\ell$ is then split into sums $S_1 + S_2 + S_r$ over the Galois conjugates of $\\alpha$; the almost-law forces the many 'small' conjugates to cluster near $1$, making $S_r$ negative with size about $c(d-4)(d-5)/2$, and the product formula yields $\\log|\\alpha| \\ge c_1(d-4)$ as well as $\\log|\\alpha| \\ge (1/(4d))\\log\\Delta_\\ell$. The volume formula for maximal arithmetic subgroups converts $\\Delta_{\\ell/k}$ into the covolume of a maximal lattice, and an argument on the tree of $PGL_2$ over the local fields controls the index of the maximal or congruence subgroup actually containing $\\Gamma$, closing the bound.","pith_inferences":["The natural sharp form of the theorem is the gap $\\log(\\operatorname{covol}(\\Gamma_1))/[k:\\mathbb{Q}]$: that is the radius at which the volume of balls in the symmetric space stops being polynomial, so a proof of Question 1.7 would make the height gap exactly match the geometric limit of the Margulis-type conclusion, and the discriminant splitting here is where such a proof would have to improve t","The same almost-law-plus-discriminant scheme should transfer to higher-rank and $S$-arithmetic groups wherever a general volume formula supplies the covolume; the main cost is a consistent bookkeeping of the parahoric subgroups, which the authors flag as the obstacle to generalization.","A concrete numerical check is available: evaluate the double-commutator almost-law word on small Zariski-dense generating sets of $PSL_2(\\mathbb{Z}[\\sqrt{-D}])$ with $D \\to \\infty$ and test whether the non-large Galois conjugates of the eigenvalue really cluster within $\\epsilon$ of $1$; if they do, $\\widehat{h}(F)$ should be observed to grow like $\\log \\operatorname{covol}(\\Gamma_1)$, and a count","Read through the analogy with classical number-theoretic height problems that the introduction draws: the covolume-dependent gap suggests effective, lattice-by-lattice lower bounds on $\\log|\\alpha|$ in terms of $d$ and $\\log\\Delta_\\ell$ that are uniform over all arithmetic lattices in a commensurability class, a closer counterpart to explicit height inequalities for algebraic numbers."],"forward_implications":["Strong arithmetic Margulis lemma (Corollary 1.6): for $\\Gamma$ arithmetic in $G$ and $x$ a point of the symmetric space, the subgroup generated by $\\{\\gamma \\in \\Gamma : d(x,\\gamma x) \\le \\epsilon_G \\log(\\operatorname{covol}(\\Gamma_1))^{1/2}\\}$ is not Zariski dense, with $\\epsilon_G$ depending only on $G$.","For lattices of large covolume the gap is genuinely of size $\\log \\operatorname{covol}(\\Gamma_1)$; when the field degree $[k:\\mathbb{Q}]$ is bounded the estimate improves to $\\widehat{h}(F) > c\\log(\\operatorname{covol}(\\Gamma_1))$, which covers in particular all non-uniform arithmetic lattices (Remark 1.5).","Finiteness of conjugacy classes of arithmetic maximal hyperbolic reflection groups in a given dimension follows without the spectral-gap input used in earlier proofs: Section 6.1 adapts an existing proof so that it depends only on the strong arithmetic Margulis lemma.","A sequence of arithmetic lattices that are either all congruence and pairwise distinct, or pairwise non-commensurable, yields Benjamini–Schramm convergence of the quotients of $\\mathbb{H}^2$ or $\\mathbb{H}^3$ to the universal cover (Theorem 6.1), proved via invariant random subgroups and the strong Margulis lemma.","Question 1.7 asks whether the denominator $[k:\\mathbb{Q}]^2$ can be lowered to $[k:\\mathbb{Q}]$; a positive answer would make Corollary 1.6 apply at the radius $\\epsilon_G \\log(\\operatorname{covol}(\\Gamma_1))$, which the authors note is optimal up to the constant because balls in non-compact symmetric spaces grow at most exponentially."],"supporting_citations":[{"why":"Classification and volume formula for maximal arithmetic subgroups of PGL2 products, which convert the field discriminant into the covolume bound.","marker":"[6]"},{"why":"Supplies the original height gap theorem whose bound Theorem 1.1 strengthens, and the height inequalities (Propositions 2.13-2.14) used to pass from the eigenvalue to the normalized height.","marker":"[7]"},{"why":"The almost-law proof of the height gap; Lemma 3.1 there is the template for Lemma 4.1, which finds a generic element in a bounded power of the generating set.","marker":"[8]"},{"why":"Uniform exponential growth for linear groups; Proposition 3.2 is invoked to guarantee the word evaluation falls outside the non-generic subvariety.","marker":"[10]"},{"why":"The recent finiteness proof for arithmetic maximal reflection groups that Section 6.1 rebuilds on the strong arithmetic Margulis lemma.","marker":"[11]"},{"why":"The arithmetic Margulis lemma and its proof strategy, which Corollary 1.6 and Section 6.2 adapt; also supplies the invariant-random-subgroup route to Benjamini-Schramm convergence.","marker":"[13]"},{"why":"The arithmetic of hyperbolic 3-manifolds reference: the splitting lemma and the theorems on fields of definition and loxodromic eigenvalues used to identify generic elements and the extension of the field of definition.","marker":"[15]"},{"why":"Existence of almost laws on compact Lie groups, the input that produces the word used in the construction of the generic element.","marker":"[20]"}],"fun_headline_variants":["Height gap in PGL2 lattices tied to log covolume","Height gap scales with covolume for PGL2 arithmetic lattices","Strong height gap: Zariski-dense sets in PGL2 get c log covol","Height in PGL2 arithmetic lattices: log covolume bound","Sharper height gap: PGL2 Zariski-dense sets scale as log covolume"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The covolume-dependent gap stands or falls on the discriminant estimate of Sections 4.3 and 5: for the generic element produced by the almost-law word, all Galois conjugates of its eigenvalue except a bounded number of 'large' ones must lie $\\epsilon$-close to $1$, making $S_r$ negative and forcing $\\log|\\alpha| \\ge c_1(d-4)$; the paper asserts the key step $\\log|\\alpha| > 10$ from the product formula without a full derivation, handles the general case with 'more crude estimates', and leaves the $PGL_2(\\mathbb{R})$ adaptation as an exercise.","fun_headline_variants_meta":{"raw":{"variants":["Height gap in PGL2 lattices tied to log covolume","Height gap scales with covolume for PGL2 arithmetic lattices","Strong height gap: Zariski-dense sets in PGL2 get c log covol","Height in PGL2 arithmetic lattices: log covolume bound","Sharper height gap: PGL2 Zariski-dense sets scale as log covolume"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000778,"raw_usage":{"total_tokens":3540,"prompt_tokens":1149,"completion_tokens":2391,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":2284}},"tokens_in":765,"tokens_out":2391,"duration_ms":17755,"temperature":1.0,"reasoning_tokens":2284,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:52:48.639791+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the double-commutator almost-law word on small Zariski-dense generating sets of $PSL_2(\\mathbb{Z}[\\sqrt{-D}])$ with $D \\to \\infty$ and compute $\\log|\\alpha|$, the degree $d$, and $\\log\\Delta_\\ell$; if for some word length the non-large Galois conjugates fail to cluster within $\\epsilon$ of $1$, or $\\log|\\alpha|$ falls below $c_1(d-4)$ while $\\langle F\\rangle$ remains Zariski dense over $\\mathbb{R}$, then the discriminant chain that links the height to $\\log\\operatorname{covol}(\\Gamma_1)$ collapses. A weaker but immediate check is to test the asserted implication $S_r \\le -c(d-4)(d-5)/2 \\Rightarrow \\log|\\alpha| \\ge c_1(d-4)$ directly on numerical discriminants, since the intermediate derivation of $\\log|\\alpha| > 10$ is not printed.","supporting_citations":[{"cited_title":"Borel , Commensurability classes and volumes of hyperbolic 3-manifolds, Ann","cited_arxiv_id":null,"evidence_quote":"Classification and volume formula for maximal arithmetic subgroups of PGL2 products, which convert the field discriminant into the covolume bound."},{"cited_title":"Breuillard , A height gap theorem for finite subsets of GLd(Q) and nonamenable sub- groups, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the original height gap theorem whose bound Theorem 1.1 strengthens, and the height inequalities (Propositions 2.13-2.14) used to pass from the eigenvalue to the normalized height."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The almost-law proof of the height gap; Lemma 3.1 there is the template for Lemma 4.1, which finds a generic element in a bounded power of the generating set."},{"cited_title":"Eskin, S","cited_arxiv_id":null,"evidence_quote":"Uniform exponential growth for linear groups; Proposition 3.2 is invoked to guarantee the word evaluation falls outside the non-generic subvariety."},{"cited_title":"Fisher and S","cited_arxiv_id":null,"evidence_quote":"The recent finiteness proof for arithmetic maximal reflection groups that Section 6.1 rebuilds on the strong arithmetic Margulis lemma."},{"cited_title":"Fraczyk, S","cited_arxiv_id":null,"evidence_quote":"The arithmetic Margulis lemma and its proof strategy, which Corollary 1.6 and Section 6.2 adapt; also supplies the invariant-random-subgroup route to Benjamini-Schramm convergence."},{"cited_title":"Maclachlan and A","cited_arxiv_id":null,"evidence_quote":"The arithmetic of hyperbolic 3-manifolds reference: the splitting lemma and the theorems on fields of definition and loxodromic eigenvalues used to identify generic elements and the extension of the field of definition."},{"cited_title":"Thom, Convergent sequences in discrete groups , Canad","cited_arxiv_id":null,"evidence_quote":"Existence of almost laws on compact Lie groups, the input that produces the word used in the construction of the generic element."}],"review_version":1}