{"id":"ded799c2-81d6-4f23-9ce4-7f111e0d9d7c","arxiv_id":"2502.02003","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For convergence groups with expanding coarse-cocycles, the author builds free subsemigroups with critical exponent arbitrarily close to the ambient critical exponent, with applications to transverse groups and Anosov semigroups.","lead":"This mathematics paper constructs, inside a wide class of infinite symmetry groups, smaller free semigroups whose exponential growth rate can be made arbitrarily close to, but strictly less than, the growth rate of the whole group. The result yields new families of Anosov semigroups in semisimple Lie groups and shows that a classical growth gap theorem for groups disappears when one allows semigroups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 3.36 asserts an unproved multiplicity bound N+1 ≥ Σ μ(γU); since μ(γU)>0 for every γ∈T, the sum can diverge while N is finite, so the strict gap δσ(T)<δσ(Γ) is not established.","rationale":"The reader's weakest assumption identifies precisely the load-bearing gap in Theorem 3.36. My independent reading agrees: the inequality N+1 ≥ Σ_{γ∈T} μ(γU) is unproved and, in natural rank-one examples, false, because μ gives positive mass to every γU while N only counts returns of U to itself. The strict inequality δσ(T)<δσ(Γ) is the core of property (4) of the main theorem, so this is a substantive gap rather than a cosmetic issue. I do not see grounds to escalate to REJECT: the construction of T, the freeness argument, the lower bound δσ(T)≥δ, and the comparability estimates are detailed and plausible, and Remark 1.4 honestly discloses the overlap with Yang's theorem. The appropriate disposition is therefore the reader's CONDITIONAL verdict: the author should prove a correct multiplicity estimate, replace the step, or weaken the statement.","tokens_in":35081,"tokens_out":5077,"duration_ms":55154,"concrete_test":"Compute the asserted inequality in the model case: Γ ⊂ PSL(2,R) a non-elementary Schottky group, T the free semigroup generated by two loxodromics via Proposition 3.2, and U an interval in ∂H² disjoint from E∪E′. For the Patterson–Sullivan measure μ of dimension δ=δ(Γ), μ(γU) ≍ e^{-δ||γ||}, so the partial sums Σ_{|γ|≤m} μ(γU) grow without bound as m→∞ while N is finite; this directly contradicts N+1 ≥ Σ μ(γU) and settles that the displayed inequality in Theorem 3.36 is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.36, after defining N = #{γ∈T : γU∩U≠∅}, the author uses the inequality N+1 ≥ Σ_{γ∈T} μ(γU). This is the only step that forces the Poincaré series of T to converge at δσ(Γ), and it is not justified. N bounds the number of elements whose image meets U; it says nothing about the total μ-measure of the images {γU}. Since U∩Λ(Γ) is nonempty and each γ is a homeomorphism preserving Λ(Γ), every γU contains a nonempty relatively open subset of the limit set. Because μ is a coarse Patterson–Sullivan measure of positive dimension, μ(γU)>0 for every γ∈T. The sum is therefore an infinite series of positive terms, and it is typically divergent at s=δσ(Γ): it is comparable to the Poincaré series of T, which diverges by Corollary 3.44. In a concrete Schottky example, N is finite while the sum over T diverges. The proof needs a genuine uniform bound on the multiplicity of the cover {γU} of Λ(Γ), or a different argument, before property (4) of Theorem 3.1 follows.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":35311,"tokens_out":9326,"duration_ms":90455,"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":[{"comment":"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.","section":"§6, proof of Theorem 6.1"}],"minor_comments":[{"comment":"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).","section":"Proof of Proposition 3.2, item (ii)"},{"comment":"In the displayed equation after the definition of the opposition involution, 'for all ι ∈ Σ' should read 'for all α ∈ Σ'.","section":"Equation (4.1)"},{"comment":"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.","section":"Lemma 3.30 and Lemma 3.39"},{"comment":"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.","section":"Lemma 3.39, proof of (3.40)"}],"recommendation":"major_revision","confidential_remarks":"The novelty is adequate: although Wenyuan Yang's earlier work already gives Theorem 1.3, the present paper's framework is different and yields the additional transverse-group and Anosov-semigroup applications. The main theorem is plausible and the missing argument in Theorem 3.36 is short, so I do not see the need for rejection. Section 6 is somewhat separate from the rest of the paper and could be trimmed, but it is a legitimate application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.","headline":"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.","tokens_in":35882,"tokens_out":5102,"would_cite":false,"duration_ms":51363,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","22E40","53C35","30F40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every convergence group with an expanding coarse-cocycle contains free subsemigroups whose critical exponent approaches its own from below.","keywords":["convergence group","expanding coarse-cocycle","critical exponent","free semigroup","Anosov semigroup","transverse subgroup","Patterson–Sullivan measure","Corlette gap theorem"],"falsifier":"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.","tokens_in":34830,"feed_emoji":"📈","tokens_out":15966,"duration_ms":133914,"temperature":0.7,"pith_summary":"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.","feed_headline":"Free semigroups nearly match the ambient critical exponent","feed_subtitle":"For hyperbolic lattices, free subsemigroups approach the full critical exponent from below, bypassing gap theorems.","key_machinery":"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 $n(R) \\geq C e^{\\delta_\\sigma(T)R}$.","core_discovery":"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 $\\delta_\\sigma(T) < \\delta_\\sigma(\\Gamma)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the GPS-system framework, shadow properties, and coarse Patterson–Sullivan measures on which the construction rests.","marker":"[3]"},{"why":"Provides the original tree construction in hyperbolic space that motivates the semigroup construction.","marker":"[2]"},{"why":"Provides the conical-limit-set counting argument adapted to obtain the linear lower bound on the counting function n(R).","marker":"[7]"},{"why":"Provides the classical technique for proving critical-exponent gaps from divergence of the Poincaré series and properness of the limit set.","marker":"[9]"},{"why":"Provides the weighted-series construction used to build the shadow-uniform measure ν on the semigroup's accumulation set.","marker":"[21]"},{"why":"Defines P_θ-Anosov semigroups, the target notion in the application to transverse subgroups.","marker":"[18]"},{"why":"Proves the theorem on C-regular quasi-isometric embeddings into symmetric spaces used in Theorem 6.1.","marker":"[10]"},{"why":"Introduces transverse (regular antipodal) groups, the class of groups to which the main application applies.","marker":"[15]"}],"fun_headline_variants":["Semigroups sneak up on the critical exponent","Free semigroups edge closer to the full exponent","Critical exponent gap bypassed by free semigroups","Approaching critical exponent from below with free semigroups","Free semigroups approach ambient critical exponent tightly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Semigroups sneak up on the critical exponent","Free semigroups edge closer to the full exponent","Critical exponent gap bypassed by free semigroups","Approaching critical exponent from below with free semigroups","Free semigroups approach ambient critical exponent tightly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00043,"raw_usage":{"total_tokens":2209,"prompt_tokens":969,"completion_tokens":1240,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":1165}},"tokens_in":585,"tokens_out":1240,"duration_ms":10254,"temperature":1.0,"reasoning_tokens":1165,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:45:32.500293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hausdorff Dimension and Klei nian Groups","cited_arxiv_id":null,"evidence_quote":"Provides the original tree construction in hyperbolic space that motivates the semigroup construction."},{"cited_title":"Mesures de Patterson–Sullivan Sur le Bor d d’un Espace Hyperbolique au Sens de Gromov","cited_arxiv_id":null,"evidence_quote":"Provides the conical-limit-set counting argument adapted to obtain the linear lower bound on the counting function n(R)."},{"cited_title":"S´ eries de Poinca r´ e des Groupes G´ eom´ etriquement Finis","cited_arxiv_id":null,"evidence_quote":"Provides the classical technique for proving critical-exponent gaps from divergence of the Poincaré series and properness of the limit set."},{"cited_title":"The Limit Set of a Fuchsian Group","cited_arxiv_id":null,"evidence_quote":"Provides the weighted-series construction used to build the shadow-uniform measure ν on the semigroup's accumulation set."},{"cited_title":"Eigenvalue Gaps for Hyperboli c Groups and Semigroups","cited_arxiv_id":null,"evidence_quote":"Defines P_θ-Anosov semigroups, the target notion in the application to transverse subgroups."},{"cited_title":"Anosov Subgroups : D ynamical and Geometric Char- acterizations","cited_arxiv_id":null,"evidence_quote":"Introduces transverse (regular antipodal) groups, the class of groups to which the main application applies."}],"review_version":1}