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Directional growth of coamenable normal subgroups: counterexamples and rigidity

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Pith's one-line read Coamenable subgroups preserve higher-rank growth exactly on opposition-symmetric directions.

desk verdict A strong paper that gives the clean higher-rank counterpart to Roblin's theorem; the counterexamples are new and the rigidity results are likely right, but the proof has a load-bearing unproved Zariski-density claim that needs fixing. read the letter →

arxiv 2606.01459 v2 pith:SE4SYQIG submitted 2026-05-31 math.GR math.DGmath.DSmath.GT

Subhadip Dey , Hee Oh , Konstantinos Tsouvalas This is my paper · ORCID
classification math.GRmath.DGmath.DSmath.GT MSC 22E4020F6520F6753C35
keywords coamenablesubgroupBorel-AnosovlimitconegrowthindicatorcriticalexponentoppositioninvolutionSchottkyRoblin'stheorem
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

Roblin's rank-one theorem says that a coamenable normal subgroup has the same critical exponent as its ambient group. The paper shows that the higher-rank directional analogue fails: for every odd n≥3 there are open families of Zariski-dense Borel-Anosov Schottky subgroups of SL_n(R) with infinite-cyclic normal subgroups whose limit cones, and hence growth indicators, are strictly smaller. Even when the limit cones coincide (in SL_3(R)), the growth indicator can differ at an interior direction. The rigidity that survives is precise: the Riemannian critical exponent is always preserved, and the growth indicator is preserved exactly on the fixed set of the opposition involution; when that involution is trivial, the full growth indicator is preserved. These three statements together are the correct higher-rank replacement for Roblin's theorem.

What carries the argument

The load-bearing object is the opposition involution i on the positive Weyl chamber, defined by µ(g^{-1}) = i(µ(g)); its fixed-point set is exactly where rigidity survives. The main technical engine is Theorem 4.5: the symmetrized inequality δ_{Γ,φ̄} ≤ δ_{N,φ} ≤ δ_{Γ,φ}, proved by defining a coarse pseudo-distance d_φ(γ_1,γ_2)=φ(µ(γ_1^{-1}γ_2)), using shadow estimates and coarse additivity of the Cartan projection for Anosov groups, and averaging a bounded logarithmic distortion over a right-invariant mean on Γ/N to obtain a character that cancels when the φ and φ∘i estimates are combined. For the counterexamples, the machinery is a perturbative ping-pong estimate (Propositions 2.2 and 2.3)

What would settle it

Exhibit a single Zariski-dense Borel-Anosov pair (Γ,N), with N coamenable, where ψ_N(v) < ψ_Γ(v) for some v in int L_N with i(v)=v; or, at the level of the engine, exhibit a Borel-Anosov Γ and positive φ for which the coarse triangle inequality d_φ(x,z) ≤ d_φ(x,y)+d_φ(y,z)+D fails — in either case Theorem 1.4 and Theorem 4.5 collapse.

Watch

Extended reading notes

Core claim

The paper's central discovery is a symmetrized coamenability inequality controlling directional critical exponents. For a Zariski-dense Borel-Anosov subgroup Γ of a connected semisimple real algebraic group and a coamenable normal subgroup N, for every linear form φ positive on L_Γ∖{0}, the paper proves δ_{Γ,φ̄} ≤ δ_{N,φ} ≤ δ_{Γ,φ}, where φ̄ = (φ+φ∘i)/2 and i is the opposition involution (the involution satisfying µ(g^{-1})=i(µ(g))). Symmetric forms therefore give equality δ_{N,φ}=δ_{Γ,φ}. From this inequality the paper derives δ_N=δ_Γ and ψ_N=ψ_Γ on the i-fixed points of a^+, using convex duality and strict concavity of growth indicators. The paper also proves this is sharp: Theorem 1.1 pro

Load-bearing premise

The proof assumes that the coarse shadow and triangle estimates for the Cartan-projection pseudo-distance hold for Borel-Anosov subgroups and that a coamenable normal subgroup is necessarily Zariski dense; if either assumption fails, the lower bound δ_{N,φ} ≥ δ_{Γ,φ̄} and the rigidity theorems do not follow.

Editorial extensions

If this is right

  • If the opposition involution is trivial (no simple factors of type A_n, D_{2n+1}, E_6), then every coamenable normal subgroup of a Zariski-dense Borel-Anosov group has the full same growth indicator and the same critical exponent as the ambient group.
  • In rank one, where i is trivial, Roblin's theorem is recovered as the special case of Theorem 1.4.
  • Amenability of the quotient does not force the limit cone to be preserved: even an infinite cyclic quotient can have strictly smaller limit cone, stably under small deformations of the representation.
  • Equal limit cones do not force equal directional growth: in SL_3(R) there are cocyclic pairs with L_N = L_Γ but ψ_N ≠ ψ_Γ at an interior direction.
  • The Riemannian critical exponent is rigid for all coamenable normal subgroups of Zariski-dense Borel-Anosov groups, a genuinely higher-rank statement with no directional caveat.

Reading between the lines

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

  • The symmetrized inequality suggests a general heuristic the paper leaves implicit: coamenability rigidity in higher rank is governed by the opposition involution, so one should expect the i-fixed locus to be the universal region where any directional invariant (limit cone, growth indicator, critical exponents of symmetric forms) is forced to coincide.
  • The perturbation estimate used for the counterexamples is open in the representation variety, so the failure of directional rigidity is not a single example but a stable, non-empty open phenomenon for Schottky-type representations in SL_n.
  • A testable extension: the proof of Theorem 4.5 relies only on coarse shadow estimates and coarse additivity, which are known for relatively Morse subgroups; if those estimates hold in that wider setting, the same rigidity should hold for cusped Hitchin or relatively Anosov subgroups — a direction the paper only remarks on.
  • The appendix's bound δ_N ≥ δ_Γ/2 for arbitrary infinite normal subgroups, together with the coamenable equality, suggests a quantitative hierarchy in which the exact constant measures how 'large' the normal subgroup is inside Γ; coamenability is the condition that upgrades the constant from 1/2 to 1.
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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

2 major / 5 minor

Summary. The paper studies higher-rank analogues of Roblin's theorem on coamenable normal subgroups of discrete groups. For Zariski-dense Borel-Anosov subgroups of connected semisimple real algebraic groups, the authors prove that the Riemannian critical exponent is preserved under coamenable normal subgroups (Theorem 1.3), and that the growth indicator is preserved exactly on the fixed-point set of the opposition involution (Theorem 1.4). They also show that full directional rigidity fails: for every odd n≥3 there is an open family of Zariski-dense Borel-Anosov Schottky subgroups of SL_n(R) with cocyclic normal subgroups having strictly smaller limit cones (Theorem 1.1), and in SL_3(R) there are cocyclic pairs with equal limit cones but different growth indicators at an interior direction (Theorem 1.5). The main technical inputs are a perturbative control of Jordan projections for free groups (Propositions 2.2–2.3) and a coamenability theorem for dual critical exponents (Theorem 4.5), proved by weighted Poincaré series and amenable averaging over Γ/N. The appendix proves a general lower bound δ_N ≥ δ_Γ/2 for arbitrary infinite normal subgroups.

Significance. If the few missing justifications are supplied, these results provide a complete and sharp higher-rank replacement for Roblin's rank-one theorem. The counterexamples are explicit, robust under perturbation, and the positive rigidity results are proved by a flexible averaging argument that is likely to extend to θ-Anosov and relatively Morse settings, as noted in Remark 4.12. The appendix's general lower bound δ_N ≥ δ_Γ/2 is also a useful standalone contribution. The paper is careful and cites the relevant literature; it does not rely on parameter fitting or circular reasoning. These are significant advances in the study of growth of higher-rank discrete subgroups.

major comments (2)
  1. [§5, first paragraph] The assertion that a coamenable normal subgroup N of a Zariski-dense Borel-Anosov subgroup Γ is 'necessarily Zariski dense' is unproved and load-bearing. It is used to apply Lemma 4.2 (to obtain L_N^i = L_Γ^i) and to ensure that L_N has non-empty interior, which is essential for the boundary-interpolation argument in Theorem 5.2. Without this premise Theorem 1.4 does not follow. A proof or precise reference is needed. A plausible argument: let H=Zar(N); then H is a normal algebraic subgroup of G, and the image of Γ in G/H is a quotient of Γ/N, hence amenable, while also being Zariski dense in G/H. Since amenable linear groups are virtually solvable and have solvable Zariski closure, G/H must be trivial. The authors should spell this out, including the treatment of finite centers and compact factors.
  2. [§4, Proposition 4.7] The extension of [23, Cor. 5.15] from the stated case C=0 to arbitrary C≥0 is asserted with the phrase 'its proof works for a general C>0' but no proof is given. If only the C=0 case is needed in the sequel (as appears to be the case for (4.7) along word geodesics), please state that and remove the unproved generalization; otherwise provide a proof. This is a missing justification in the foundational estimates used for Theorem 4.5.
minor comments (5)
  1. [§4, before Lemma 4.4] Typo: 'W will use' should be 'We will use'.
  2. [§4, Lemma 4.9] The notation O_R(1) is used in the proof without defining it; it presumably means a constant depending only on R. Please clarify to avoid confusion.
  3. [§6, Lemma 6.10] The proof is somewhat compressed. In particular, the statement that 'by the convergence dynamics of the Anosov subgroup ∆, all but finitely many elements g∈∆ satisfy gX⊂intY' should be justified in more detail, and the simultaneous choice of ∆' satisfying both the ping-pong condition and the Zariski-density condition (2) needs a clearer argument. The lemma is important for the L_N=L_Γ refinement in Theorem 6.9.
  4. [§3, equation (3.3)] The Borel-Anosov condition for the representation ρ_κ is quoted from [22]. Since [22] is a preprint, please ensure the precise statement and hypotheses are publicly available and that the condition (3.3) indeed matches [22].
  5. [§5, Theorem 5.1] The statement 'L_Γ − {0} ⊂ int a^+' is attributed to [15]; it may depend on the Borel-Anosov assumption and on a choice of positive Weyl chamber. Please make the hypotheses explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained; the flagged items are unproved premises or extensions, not reductions to inputs.

full rationale

The main rigidity results (Theorems 1.3 and 1.4) are derived from Theorem 4.5, whose proof is carried out directly via shadow estimates, a weighted convolution inequality, and amenable averaging over Γ/N; it does not presuppose the equality δ_{Γ,φ̄} = δ_{N,φ} that it proves. The positive results therefore do not reduce to their own conclusion. The paper does import several earlier results, including [9, Thm 4.1] for the coarse triangle inequality and [23, Cor 5.15] for Cartan projection estimates; these are general coarse-geometric tools with stated hypotheses that do not include the target theorem, so they are independent support rather than circular self-citation. The construction in Theorem 1.1 is explicit: it tracks the middle Jordan coordinate λ_{d+1}(ρ_κ(γ)) = κχ(γ), and the separation of limit cones follows from a perturbation estimate, not from fitting a parameter to the predicted quantity. No fitted input is renamed as a prediction. Two rigor gaps are present but are not circularity: (i) Proposition 4.7 states the extension of [23, Cor 5.15] from C=0 to arbitrary C with 'its proof works' and no proof; (ii) Section 5 asserts without proof that a coamenable normal subgroup of a Zariski-dense Borel–Anosov group is 'necessarily Zariski dense,' a load-bearing structural premise for Lemma 4.2 and for int(L_N) ≠ Ø. Both are correctness/robustness concerns, not instances where an equation reduces to its own input by construction. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Construction parameters κ, m, r, and the finite-index m are existential choices satisfying explicit inequalities ((3.3), (2.1), (6.5)); they are not fitted values, and the theorems hold for all sufficiently small or large choices. No empirical data or fitted constants enter the argument.

assumptions (6)
  • domain assumption Benoist's limit-cone theorem: for a Zariski-dense discrete subgroup Γ<G, L_Γ is the smallest closed cone containing the Jordan projections of elements of Γ (Theorem 4.1, cited [4]).
    Used to identify L_{σ(N)} in Theorem 1.1 and to prove L^i_N = L^i_Γ in Lemma 4.2.
  • domain assumption Borel-Anosov subgroups are word hyperbolic, their orbit maps are quasi-isometric embeddings, their limit cones satisfy L_Γ−{0}⊂int a^+, and their growth indicators are strictly concave on int L_Γ (Theorem 5.1, cited [15,25,29]).
    Essential for Lemma 4.6, the tangent-form argument in Theorem 5.2, and uniqueness of maximizing rays.
  • domain assumption Quint's variational formula δ_{Γ,φ} = sup_{v∈L_Γ−{0}} ψ_Γ(v)/φ(v) and the tangent-form criterion of [20, Cor 4.6] hold for these subgroups (Lemma 4.4).
    Converts dual-critical-exponent equality into growth-indicator identity in Theorems 1.4 and 6.9.
  • domain assumption The coarse triangle inequality and shadow estimates for the pseudo-distance d_φ hold for Borel-Anosov groups (Propositions 4.7–4.8, from [23, Cor 5.15] and [9, Thm 4.1]).
    Underpins the weighted convolution Lemma 4.10 and the amenable-averaging Lemma 4.11; Proposition 4.7's extension to C>0 is asserted without proof.
  • domain assumption An amenable quotient Γ/N of a Zariski-dense non-elementary Borel-Anosov group forces N to be Zariski dense in G.
    Invoked in Section 5 without proof; needed for Lemma 4.2 and the limit-cone equalities.
  • domain assumption Openness of Zariski density and of the Borel-Anosov property under perturbation, together with Schottky combination theorems (cited [8,16,22,31]).
    Used to produce the non-empty open family in Theorem 1.1 and the generic choice of c in Theorem 6.9.

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Pith. "Pith review of Directional growth of coamenable normal subgroups: counterexamples and rigidity." pith.science (2026). https://pith.science/paper/SE4SYQIG

@misc{pith2026260601459,
  author       = {Pith},
  title        = {Pith review of: Directional growth of coamenable normal subgroups: counterexamples and rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SE4SYQIG}},
  note         = {Machine review of arXiv:2606.01459}
}
abstract

Roblin's theorem asserts that, in rank one, a coamenable normal subgroup has the same critical exponent as its ambient group. Natural higher-rank analogues would predict that coamenability preserves the limit cone and the growth indicator. We show that both assertions fail, even when the quotient is infinite cyclic. For every odd integer $n\geq 3$, we construct a nonempty open family of Zariski-dense Borel--Anosov Schottky subgroups of $\mathrm{SL}_n(\mathbb R)$ admitting cocyclic normal subgroups with strictly smaller limit cones. Moreover, in $\mathrm{SL}_3(\mathbb R)$, we construct cocyclic pairs with the same limit cone but distinct growth indicators at an interior direction. We then identify the precise rigidity that survives. Let $\Gamma$ be a Zariski-dense Borel--Anosov subgroup of a connected semisimple real algebraic group, and let $N\lhd\Gamma$ be coamenable. Then the growth indicators of $N$ and $\Gamma$ agree on the fixed-point set of the opposition involution, and their Riemannian critical exponents are equal. The examples with equal limit cones show that the restriction to opposition-invariant directions is sharp.

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