Free Semigroups of Large Critical Exponent

Aleksander Skenderi

REVIEW 1 major objections 4 minor 25 references

Free Semigroups of Large Critical Exponent

T0 review · 1 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Every convergence group with an expanding coarse-cocycle contains free subsemigroups whose critical exponent approaches its own from below.

desk verdict A genuinely new construction of large-exponent free subsemigroups in convergence groups, but the strict gap in Theorem 3.36 rests on an unproved and likely false multiplicity bound. read the letter →

arxiv 2502.02003 v3 pith:MHOZQN2D submitted 2025-02-04 math.GR math.DSmath.GT

Aleksander Skenderi This is my paper · ORCID

classification math.GRmath.DSmath.GT MSC 20F6522E4053C3530F40

keywords convergencegroupexpandingcoarse-cocyclecriticalexponentfreesemigroupAnosovtransversesubgroupPatterson–SullivanmeasureCorlettegaptheorem

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

Starting from any non-elementary convergence group equipped with an expanding coarse-cocycle, this paper constructs free finitely generated subsemigroups whose critical exponent is arbitrarily close to, yet always strictly smaller than, the critical exponent of the ambient group. These Bishop–Jones semigroups are built by choosing a finite generating set whose shadows are disjoint and nest along words, and the construction forces a linear comparison between word length and any expanding cocycle magnitude. A key consequence is that for transverse subgroups of semisimple Lie groups, there are free Anosov subsemigroups approximating the ambient critical exponent from below. In the special case of lattices in quaternionic and octonionic hyperbolic spaces, this shows that the classical gap in critical exponents for groups (Corlette's gap) does not exist for semigroups.

What carries the argument

The central machinery is the Bishop–Jones semigroup $T = \bigcup_{m\geq 0} S^m$, generated by a finite set $S$ constructed in Proposition 3.2. The generators are chosen from the level set $A_{\sigma,n} = \{\gamma \in \Gamma : n \leq ||\gamma||_\sigma < n+1\}$ inside a ball $B_{t_0/2}(x)$ around a limit point $x$, after right multiplication by elements of a finite uniform-loxodromic set $F$. The construction's four load-bearing properties are: (1) the shadows $S_{t_0/4}(\eta)$ of children $\eta \in \gamma\cdot S$ lie inside $S_{t_0/2}(\gamma)$; (2) these child shadows are pairwise disjoint; (3) $||\eta^{-1}\gamma||_\sigma \leq D_0$ for each child; and (4) the $\delta$-weighted sum $\sum_{\eta\in\gamma\cdot S} e^{-\delta||\eta||_\sigma} \geq e^{-\delta||\gamma||_\sigma}$. From these, the paper deduces that $T$ is free, that its accumulation set $E$ is uniformly conical, that $E\cup E'$ is a proper subset of the ambient limit set, and that the Poincaré series of $T$ diverges at $\delta_\sigma(T)$. A shadow-uniform measure $\nu$ on $E$, built from a Patterson-style weighted series, provides the upper bound $\nu(S_{t_0/2}(\gamma)) \leq C_1 e^{-\delta||\gamma||_\sigma}$ that feeds into the counting estimate $n(R) \geq C e^{\delta_\sigma(T)R}$.

What would settle it

Implement Proposition 3.2 for a classical Schottky group acting on the Riemann sphere, letting $\delta_m$ approach $\delta_\sigma(\Gamma)$, and compute the critical exponent of the resulting free semigroup; if any of these exponents equals $\delta_\sigma(\Gamma)$, or if the asserted inequality $N+1 \geq \sum_{\gamma\in T} \mu(\gamma U)$ fails for the open set $U$ of Lemma 3.45, Theorem 3.1(4) is refuted.

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Extended reading notes

Core claim

The paper's central claim, Theorem 3.1, is that for every $0 < \delta < \delta_\sigma(\Gamma)$ there exists a free finitely generated subsemigroup $T = T_\delta \subset \Gamma$, with finite generating set $S$, such that $\delta_\sigma(T) \geq \delta$ and $\delta_\sigma(T) < \delta_\sigma(\Gamma)$. The same theorem gives three quantitative companions: word length in $S$ is linearly equivalent to any expanding coarse-cocycle magnitude on $T$; the $\sigma$-critical exponent of $T$ is comparable to $\log(\#S)$; and the critical exponent of $T$ is finite even when $\delta_\sigma(\Gamma)$ is infinite. The construction works by selecting a finite set $S$ from a level set of the cocycle near a limit point, in such a way that the shadows of its elements are pairwise disjoint, the shadow of each child lies inside the parent's shadow, the inverse magnitude of each child relative to its parent is uniformly bounded, and the $\delta$-weighted Poincaré sum over children is at least the parent's contribution. These four properties force the semigroup to be free, give the lower bound $\delta_\sigma(T) \geq \delta$, and support the divergence of the $\sigma$-Poincaré series of $T$ at $\delta_\sigma(T)$, which is the key to proving the strict gap $\delta_\sigma(T) < \delta_\sigma(\Gamma)$.

Load-bearing premise

The proof of the strict gap $\delta_\sigma(T) < \delta_\sigma(\Gamma)$ in Theorem 3.36 depends on the unproved inequality $N+1 \geq \sum_{\gamma\in T} \mu(\gamma U)$ for a small open set $U$ disjoint from $E\cup E'$, where $\mu$ is a coarse Patterson–Sullivan measure for $\Gamma$; this asserts that the number of returns of $U$ under the semigroup dominates the total $\mu$-measure of the shadows $\gamma U$, thereby controlling overlap multiplicity.

Editorial extensions

If this is right

For any non-elementary $P_\theta$-transverse subgroup $\Gamma$ of a semisimple Lie group and any $\varphi \in \mathfrak{a}^*_\theta$ positive on the cone, there are free $P_\theta$-Anosov subsemigroups $\Gamma_n \subset \Gamma$ with $\delta_\varphi(\Gamma_n) < \delta_\varphi(\Gamma)$ and $\delta_\varphi(\Gamma_n) \to \delta_\varphi(\Gamma)$ (Theorem 5.1).
Lattices in $\mathrm{Sp}(n,1)$ and $F^{-20}_4$ contain free subsemigroups with critical exponent approaching $4n+2$ and 22, respectively, from below; hence the Corlette gap theorem, which forces discrete groups to have exponent either equal to the lattice value or at most $4n$ (resp. 16), has no semigroup analog (Theorem 5.2).
Bishop–Jones semigroups map quasi-isometrically into the symmetric space of $G$, and for certain linear functionals the displacement $d_\varphi$ satisfies a coarse triangle inequality on the embedded semigroup (Theorem 6.1).
On any Bishop–Jones semigroup, word length and any expanding coarse-cocycle are within bounded additive error of being linear functions of each other, so the critical exponent of $T$ is finite and comparable to $\log(\#S)$.
The accumulation set of $T$ is contained in the uniformly conical limit set of $\Gamma$, and $E \cup E'$ is a proper closed subset of $\Lambda(\Gamma)$, which is what forces the strict inequality in the finite critical exponent case.

Reading between the lines

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

Because the Busemann cocycle on any proper geodesic Gromov hyperbolic space forms a GPS system, the same construction should produce free subsemigroups with critical exponent arbitrarily close to the group's for every non-elementary group of hyperbolic isometries, giving an independent route to results obtained by contracting-element methods.
The paper notes that the group generated by a Bishop–Jones generating set $S$ in a rank-one lattice is itself a lattice; this suggests the critical exponent is sensitive to the presence of inverses, and raises the question whether a Bishop–Jones semigroup can be quasi-isometric to its ambient group while having a strictly smaller critical exponent.
The shadow-uniform measure $\nu$ on the accumulation set $E$ is constructed to behave like a Patterson–Sullivan measure for the semigroup; it may be useful for counting semigroup orbits or for proving mixing statements for the one-sided action.
Relaxing the disjoint-shadow condition could yield free subsemigroups with even larger critical exponents, or non-free subsemigroups, and might sharpen the rate at which $\delta_\sigma(T)$ approaches $\delta_\sigma(\Gamma)$ as $\delta$ varies.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper delivers a genuinely new construction: Bishop–Jones semigroups in any convergence group with an expanding coarse-cocycle, with critical exponents approaching the ambient exponent from below. The application to transverse groups (Theorem 5.1) producing free Anosov subsemigroups is new and interesting, and the Corlette-gap corollary is a nice bonus, with the overlap with Yang's theorem fully acknowledged. The writing is clear, and the inductive construction in Proposition 3.2 is careful and appears sound. Properties (1)–(3) and (5) of Theorem 3.1 follow from that construction without serious issues.

The soft spot is the proof of property (4) in Theorem 3.36. The author defines N = #{γ ∈ T : γU ∩ U ≠ ∅} and then asserts N+1 ≥ Σ_{γ∈T} μ(γU). That inequality is neither proved nor, on its face, justified. N only counts how many translates of U meet U; it says nothing about the total μ-measure of all the images {γU} on the limit set. Since each γU is a relatively open subset of Λ(Γ) of positive μ-measure, the sum is an infinite series of positive terms. Under the contradiction assumption δσ(T)=δσ(Γ), the lower bound in the same chain of inequalities shows this sum is comparable to the divergent Poincaré series of T at δσ(Γ). So the asserted inequality would force that series to converge, which contradicts Corollary 3.44. The inequality is load-bearing: it is the only step that produces the contradiction. A real multiplicity bound for the cover {γU}, or a different argument, is needed before property (4) follows.

The rest of the paper—the free-semigroup construction, the coarse-cocycle estimates, the transverse-group application, and the quasi-isometric embedding section—has independent value and is honestly presented. This is a paper for researchers in geometric group theory, Anosov representations, and Patterson–Sullivan theory. It deserves a serious referee, but the referee should insist on repairing or replacing the argument in Theorem 3.36.

Referee Report

1 major / 4 minor

Summary. The paper introduces 'Bishop–Jones semigroups' for a non-elementary convergence group equipped with an expanding coarse-cocycle. The main theorem (Theorem 3.1) asserts that for every 0 < δ < δ_σ(Γ) there is a free finitely generated subsemigroup T ⊂ Γ whose σ-critical exponent is at least δ and strictly smaller than δ_σ(Γ), and that satisfies additional linear-growth and quasi-isometric-type estimates. The construction is carried out in Proposition 3.2 via a shadow-tree argument in the spirit of Bishop–Jones. The paper then applies the main theorem to P_θ-transverse subgroups of semisimple Lie groups, obtaining free P_θ-Anosov subsemigroups with critical exponents approximating the ambient critical exponent from below (Theorem 5.1), and consequently showing that Corlette's gap theorem for lattices in Sp(n,1) and F_4^{-20} has no analogue for subsemigroups (Theorem 5.2). A further application shows that these semigroups admit C-regular quasi-isometric embeddings into the symmetric space (Theorem 6.1).

Significance. If the main theorem is correct, it is a substantial and elegant result: it gives a general mechanism, valid for all expanding coarse-cocycles, for producing free subsemigroups whose critical exponent is arbitrarily close to the ambient one while remaining strictly below it. The applications to transverse groups and to Kassel–Potrie Anosov semigroups are natural and significant, and the consequence for Corlette's gap theorem is striking. The paper is carefully written and makes good use of the GPS framework of Blayac–Canary–Zhu–Zimmer. The main proof is detailed and the shadow arguments are coherent. However, one step in the proof of the strict-inequality property is asserted without justification; it is true and easily repairable, but it is load-bearing and should be fixed. The Section 6 application also needs a small clarification about the geodesic metric space to which the Dey–Kim–Oh theorem is applied.

major comments (1)

[§6, proof of Theorem 6.1] The proof says 'We view the Bishop–Jones semigroup T as a geodesic metric space by equipping it with its tree metric', but the vertex set of a tree with the induced path metric is not itself a geodesic metric space. Theorem 6.4 requires a geodesic metric space. The fix is straightforward: apply Theorem 6.4 to the Cayley graph of T (a rooted tree with edges labelled by the generating set S) and extend the map f to edges by geodesic segments in X, or state explicitly that T is identified with the vertex set of its geodesic Cayley graph. As written, the hypothesis of Theorem 6.4 is not literally satisfied.

minor comments (4)

[Proof of Proposition 3.2, item (ii)] In the paragraph after defining S, the text writes 'η=γ f_i for some γ ∈ P_i ∩ A_{σ,n} ∩ B(x,ǫ)', but the radius ǫ is undefined here; it should be B_{t0/2}(x).
[Equation (4.1)] In the displayed equation after the definition of the opposition involution, 'for all ι ∈ Σ' should read 'for all α ∈ Σ'.
[Lemma 3.30 and Lemma 3.39] Both proofs use the inclusion S_{t0/2}(α) ⊂ S_{t0/4}(α), which follows from the definition of shadow because a larger ball gives a smaller complement. This inclusion is not stated; adding it once would make the arguments easier to follow.
[Lemma 3.39, proof of (3.40)] The proof uses the implication S_{t0/2}(g_n) ⊂ S_{t0/2}(η) for g_n descending from η ∈ S^n; this follows from S_{t0/2}(g_n) ⊂ S_{t0/4}(g_n) and repeated application of property (1) of Proposition 3.2, but the reader must supply these steps.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the semigroup construction and critical-exponent gap are proved from external GPS/PS theory and from direct tree-growth estimates, not from the theorem being proved.

full rationale

The paper's main derivation is not circular. Theorem 3.1 constructs a finite generating set S so that the delta-Poincare series of the semigroup diverges: Property (4) of Proposition 3.2 directly enforces sum_{eta in gamma S} e^{-delta||eta||_sigma} >= e^{-delta||gamma||_sigma}, and Corollary 3.27 then gives delta_sigma(T) >= delta. This is a lower bound built into the construction, not an output fitted to itself. The upper bound delta_sigma(T) <= B1 log(#S) follows from the word-length comparison in Lemma 3.29 and the exact growth count for a free semigroup in Proposition 3.34, neither of which imports the conclusion. The strict gap delta_sigma(T) < delta_sigma(Gamma) in Theorem 3.36 is a contradiction argument: assuming equality, the ambient coarse Patterson-Sullivan measure and the divergence of T's Poincare series at its critical exponent (Corollary 3.44) would force the infinite sum sum_{gamma in T} mu(gamma U) to diverge, contradicting the finiteness of N. Every ingredient here is either proved inside the paper or imported as an external theorem from Blayac-Canary-Zhu-Zimmer, Dey-Kim-Oh, Corlette, or standard Lie theory; the author has no load-bearing self-citation. The skeptical concern about the unproved multiplicity bound N+1 >= sum mu(gamma U) is a potential correctness gap in the proof of Theorem 3.36, but it is not circularity: the inequality is not equivalent to the theorem's statement, and no fitted parameter or definitional identity makes the conclusion equal to an input. For these reasons the circularity score is 0.

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

The central claim rests on a substantial external framework (GPS systems, coarse cocycles, transverse groups) that is assumed as background. The paper's own contribution is the construction of the semigroup inside that framework; it does not introduce new postulates or fitted parameters. The main unverified input is the completeness of the proof of Theorem 3.36.

assumptions (6)

domain assumption Expanding coarse-cocycles satisfy the growth and quasi-multiplicativity estimates of Proposition 2.5 (parts (1)-(3)) from [3].
These estimates are imported from Blayac-Canary-Zhu-Zimmer and are used throughout Proposition 3.2 and Lemmas 3.29, 3.39.
standard math For a non-elementary convergence group, the limit set is perfect and minimal, and shadows satisfy the diameter/location estimates of Proposition 2.10 from [3].
Used in Lemmas 3.7, 3.11, 3.31, 3.33 to control limits of semigroup elements.
domain assumption For a P_theta-transverse group, the partial Iwasawa cocycle gives a continuous GPS system (Proposition 4.4 of [4]).
This is the bridge from the abstract convergence-group theorem to transverse subgroups of Lie groups.
domain assumption Corlette's gap theorem (Theorem 1.2) states the critical exponent values for lattices in Sp(n,1) and F4^-20.
Used in Theorem 5.2 to identify delta(Gamma)=4n+2 and 22.
domain assumption Dey-Kim-Oh Theorem 6.4 provides the coarse triangle inequality for C-regular quasi-isometric embeddings into symmetric spaces.
The quasi-isometric embedding section relies on this external result.
domain assumption The Busemann cocycle on a rank-one symmetric space forms a continuous GPS system (Example 2.8).
Used in the proof of Theorem 5.2 for H^n_H.

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Pith. "Pith review of Free Semigroups of Large Critical Exponent." pith.science (2026). https://pith.science/paper/MHOZQN2D

@misc{pith2026250202003,
  author       = {Pith},
  title        = {Pith review of: Free Semigroups of Large Critical Exponent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHOZQN2D}},
  note         = {Machine review of arXiv:2502.02003}
}

abstract

For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call Bishop--Jones semigroups, of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any non-elementary transverse subgroup $\Gamma$ of a semisimple Lie group $G$, there exist finitely generated free Anosov subsemigroups in the sense of Kassel--Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit quasi-isometric embeddings into the symmetric space X of G with certain additional coarse-geometric properties.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Semigroups Co ntaining Proximal Linear Maps

H. Abels, G.A. Margulis, and G.A. Soifer, “Semigroups Co ntaining Proximal Linear Maps”, Israel J. Math. , 91, p.1-30, (1995)

work page 1995

[2] [2]

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C.J. Bishop and P.W. Jones, “Hausdorﬀ Dimension and Klei nian Groups”, Acta Math. , 179 (1), p.1-39, (1997)

work page 1997

[3] [3]

Patterso n–Sullivan Theory for Coarse Cocycles

P.-L. Blayac, R. Canary, F. Zhu, and A. Zimmer, “Patterso n–Sullivan Theory for Coarse Cocycles”, arXiv:2404.09713, (2024)

arXiv 2024

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P.-L. Blayac, R. Canary, F. Zhu, and A. Zimmer, “Counting , Mixing and Equidistribution for GPS Systems with Applications to Relatively Anosov Groups” , arXiv:2404.09718, (2024). 40 ALEKSANDER SKENDERI

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work page 2024

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Mesures de Patterson–Sullivan Sur le Bor d d’un Espace Hyperbolique au Sens de Gromov

M. Coornaert, “Mesures de Patterson–Sullivan Sur le Bor d d’un Espace Hyperbolique au Sens de Gromov”, Paciﬁc J. Math. , 159 (2), p.241-270, (1993)

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K. Corlette, “Hausdorﬀ dimensions of limit sets I”, Invent. Math. , 102, p.521-542, (1990)

work page 1990