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Zariski density of discrete subgroups via critical exponents and unitary representations

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Pith's one-line read 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)…

desk verdict 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. read the letter →

arxiv 2608.04966 v1 pith:MLFSAWGN submitted 2026-08-05 math.GR math.DSmath.GTmath.RT

Aleksander Skenderi This is my paper · ORCID
classification math.GRmath.DSmath.GTmath.RT MSC 22E4022E46
keywords ZariskidensitycriticalexponentsemisimpleLiegroupsunitaryrepresentationstemperednesslatticetheoremlimitconerealrankone
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

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 $\Gammah_{\mathrm{vol}}(X)-\epsilon(G)$ 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 $F_4^{-20}$.

What carries the argument

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 $L^2$.

What would settle it

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 $\epsilon>0$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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).

Reading between the lines

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

  • 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.
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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 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).

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 (3)
  1. [Section 4, Theorem 4.10(a) and (b)] 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.
  2. [Section 4, proof of Proposition 4.2] 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.
  3. [Section 4, Proposition 4.1 and proof of Theorem 4.4] 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.
minor comments (3)
  1. [Section 4, proof of Proposition 4.2] 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.
  2. [Section 4, proof of Theorem 4.4] 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.
  3. [Introduction and Remark 4.11] 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).

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the sole self-citation [28] is a non-load-bearing pointer to Quint's result.

full rationale

The central bound in Theorem 4.4 is derived, not assumed: for a non-Zariski-dense Gamma, the proof splits into the reductive case (delta(Gamma) <= hvol(Y) for a proper symmetric subspace Y) and the parabolic case (delta(Gamma) <= (1 - 1/p_{G/Q}) hvol(X) via the paper's Prop. 3.6, Thm. 3.9, Cor. 3.10 and Quint's maximizing vector w). No parameter is fitted to Gamma, and no quantity in the bound is defined in terms of Zariski density, so the conclusion does not reduce to its input. The finiteness of p_{G/Q} in (4.6) rests on Prop. 4.1, quoted from Einsiedler-Margulis-Venkatesh and Moore, and on Lemma 2.7 from Samei-Wiersma; these are external results whose validity is a correctness concern, not circularity. Borel's density theorem is used only as a benchmark and is recovered as a lattice case, not used as an input. The one self-citation, [28], appears only as a 'for more details' pointer attached to Quint [26] for the existence of w with psi_Gamma(w) = delta(Gamma); that fact is independently sourced to Quint, so [28] is not load-bearing. Remark 4.11 openly leaves optimality in the F^{-20}_4 case undecided, which is additional evidence that the quantitative claims are not forced by construction. Overall, no circular step is present in the derivation chain.

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

The paper introduces no free parameters and no new entities. It depends on recent external results from Benoist-Liang, on the EMV/Moore spectral gap, and on standard facts about critical exponents and growth indicators. The main derivation is not circular but is heavily borrower-based, so confidence in the final epsilon depends on the cited black boxes.

assumptions (5)
  • domain assumption Benoist-Liang identity delta_{G/H} = theta_{G/H} = 1 - 1/p_{G/H} for closed subgroups H (Theorem 3.9).
    Invoked in the proof of Theorem 4.4 after (4.7) to bound delta(Gamma) via delta_{G/Q}. Not proved in this paper; external result [9].
  • domain assumption Proposition 4.1: for G simple non-compact and H a proper closed connected subgroup, L^2(G/H) is almost L^{2m} for some m (EMV/Moore).
    Black box used to prove finiteness of p_{G/Q} for parabolic subgroups. No proof or explicit m is given here.
  • domain assumption Quint's growth indicator: there is a unit vector w in a+ with psi_Gamma(w) = delta(Gamma) and delta(Gamma) = max psi_Gamma.
    Used to convert the bound psi_Gamma(v) <= (1 - 1/p_{G/Q}) 2 rho(v) into delta(Gamma) <= (1 - 1/p_{G/Q}) h_vol(X). Cited to [26] and [28].
  • domain assumption For a reductive subgroup H, delta(Gamma) <= h_vol(Y) for Gamma < H, with equality for lattices in H.
    Used in Case 1 of Theorem 4.4; standard but not proved in this paper.
  • standard math There are finitely many isometry classes of proper totally geodesic symmetric subspaces of X.
    Needed for the maximum in (4.5) to be strictly less than h_vol(X).

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Cite this review

Pith. "Pith review of Zariski density of discrete subgroups via critical exponents and unitary representations." pith.science (2026). https://pith.science/paper/MLFSAWGN

@misc{pith2026260804966,
  author       = {Pith},
  title        = {Pith review of: Zariski density of discrete subgroups via critical exponents and unitary representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLFSAWGN}},
  note         = {Machine review of arXiv:2608.04966}
}
abstract

Let $G$ be a connected real semisimple linear algebraic group with finite center and no compact factors, let $X$ denote its associated Riemannian symmetric space, and let $h_{\mathrm{vol}}(X)$ be the volume growth entropy of $X$. We show that there exists an $\epsilon = \epsilon(G) > 0$ so that the following holds: if $\Gamma < G$ is a discrete subgroup with critical exponent greater than $h_{\mathrm{vol}}(X) - \epsilon$, then $\Gamma$ is Zariski dense in $G$. If $G$ is further assumed to be isomorphic to one of the isometry groups of the real, complex, or quaternionic hyperbolic spaces, we determine the smallest possible value of the critical exponent to guarantee Zariski density of the associated subgroup (in other words, the largest possible value of $\epsilon = \epsilon(G)$). In the setting of discrete subgroups of real semisimple Lie groups with no compact factors, this generalizes Borel's Density Theorem for lattices. The key ingredient is input from the theory of unitary representations, particularly the recent work of Benoist--Liang (which was inspired by earlier work of Benoist--Kobayashi) on general temperedness criteria for the quasi-regular representation of $L^2(G/H)$ for any closed subgroup $H$ of $G$.

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