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A strong height gap theorem for $PGL_2$

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.22266 v1 pith:JGX77OR5 submitted 2025-07-29 math.GR

Mikhail Belolipetsky , Sebastian Hurtado This is my paper · ORCID
classification math.GR MSC 22E4011F06
keywords heightgaptheoremarithmeticlatticescovolumePGL2ZariskidensesubgroupsalmostlawsdiscriminantestimatesMargulislemma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [Section 4.3, Eqs. (3)–(5)] 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.
  2. [Section 4.5] 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.
  3. [Section 5, Eqs. (12)–(14)] 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.
minor comments (3)
  1. [Throughout] 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.
  2. [Section 4.3] 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].
  3. [Proof of Corollary 1.6] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the covolume-dependent height bound is obtained from fresh word/discriminant estimates and Borel's volume formula; the deferred bounded-degree cases are completeness gaps, not self-referential reasoning.

full rationale

The derivation chain is: choose an almost-law word w (via Thom/Lindenstrauss and the method of [8]), find a generic value W = w(A0,B0) in Γ^(2) (Lemma 4.1), estimate the discriminant Δ_ℓ of the eigenvalue field in terms of conjugates of α (Section 4.3), convert log|α| to the normalized height of F through the definition ĥ(F) ≥ h(α)/|w|n_w (Eq. (9)), and compare the discriminant, via Borel's volume formula, with covol(Γ1). None of these ingredients is the theorem being proved: the desired lower bound ĥ(F) > c log(covol)/[k:Q]^2 is not assumed to construct W or to bound Δ_ℓ; it is the output of the chain. The cited papers [8] and [13] supply the almost-law method and the pattern of the arithmetic Margulis lemma, respectively; they do not contain the covolume-dependent gap, and [8]'s height-gap theorem is used as a method, not as a premise. Thus the proof does not reduce to its own conclusion by construction, by fitted parameters, or by a self-citation chain. Two genuine issues appear, but they are not circularity. Section 4.3 says 'the case of bounded degree is much simpler and we will deal with it later on', and Section 5 repeats 'When the degree is bounded the proof simplifies significantly as we will see later on', yet no bounded-degree argument is supplied; for Bianchi groups (d = 4) the set α5,...,αd is empty, so inequalities (3) and (5) rest on an unproved case. Likewise, 'By choosing ε > 0 sufficiently small we can assume that |α1| > 10' is asserted without a full derivation. These are gaps in justification, not reductions of the conclusion to the assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The proof relies on standard theorems (Borel volume formula, uniform exponential growth, almost laws, quaternion algebra splitting, Borel density) and on a delicate discriminant estimate that is partially sketched.

assumptions (5)
  • standard math Borel's volume formula for covolumes of maximal arithmetic lattices in PGL2(R)^a × PGL2(C)^b (formula (1)).
    Used in Section 4.4 to relate the discriminant Δ_ℓ to covol(Γ0). Cited to [6].
  • standard math Uniform exponential growth for linear groups ([10, Proposition 3.2]) used in Lemma 4.1 to find generic elements in F^{n_w}.
    Key input for Lemma 4.1; established result.
  • standard math Existence of ε-almost laws on compact Lie groups (Thom [20], attributed to Lindenstrauss).
    Used in Section 4.2 to choose the word w0 making conjugates of a generic element close to 1.
  • standard math Splitting criterion for quaternion algebras over number fields ([15, Lemma 12.2.1, Theorems 12.2.3]) used in Lemma 3.3.
    Establishes that ℓ = Q(α) splits the quaternion algebra A(k), linking the generic eigenvalue to arithmetic invariants.
  • domain assumption Borel density: arithmetic lattices are Zariski dense, so Γ×Γ is Zariski dense in G×G.
    Used implicitly in Lemma 4.1 to ensure the subvariety Y is proper and its complement is hit by F^{n_w}×F^{n_w}.

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Pith. "Pith review of A strong height gap theorem for $PGL_2$." pith.science (2026). https://pith.science/paper/JGX77OR5

@misc{pith2026250722266,
  author       = {Pith},
  title        = {Pith review of: A strong height gap theorem for $PGL_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JGX77OR5}},
  note         = {Machine review of arXiv:2507.22266}
}
abstract

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was given later by Chen, Hurtado and Lee. In this paper we show that when the set $F$ is contained in a maximal arithmetic subgroup $\Gamma$ of $G = PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$, $a+b \ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log(covol(\Gamma))$, the function of the covolume of $\Gamma$. This result strengthens the theorem for the lattices of large covolume and has various applications including a strong version of the arithmetic Margulis lemma for $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$.

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