{"id":"4636ecdf-4362-4fe9-8b80-1d54ebef68d2","arxiv_id":"2607.25578","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"WPD elements are exponentially generic for every finite generating set of an acylindrically hyperbolic group, yielding growth tightness and cogrowth tightness.","lead":"This paper proves that in acylindrically hyperbolic groups—a broad class including mapping class groups and Out(F_n)—the 'rotating' (WPD) elements are exponentially generic in every Cayley graph. It settles open questions about growth tightness and genericity with respect to arbitrary finite generating sets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem A needs three independent WPD elements (Section 4) but the theorem assumes only two; the two-to-three passage is never justified.","rationale":"The reader identified the same weakest assumption: the proof uses three independent WPD elements while Theorem A assumes two, with no justification of the passage. This is indeed the most load-bearing concern about the central claim because the projection-complex and insertion machinery collapses without a three-element F. I checked the surrounding arguments — the anchored length estimate (Proposition 3.4), the counting lemma (Lemma 4.9), the guard decomposition (Section 5), and the genericity conclusion — and found no other substantive gap; the counting algebra and the WPD-to-loxodromic transfer (Lemmas 5.8, 2.37) appear sound. The two-to-three issue is real but likely easily repaired by conjugating one element by the other, so it does not warrant rejection; it does warrant a conditional verdict pending the missing justification. Hence I agree with the reader's CONDITIONAL verdict and see no need to change it.","tokens_in":46508,"tokens_out":15065,"duration_ms":136362,"concrete_test":"Verify the two-to-three reduction. Concretely: take h1,h2 independent strongly contracting WPD elements and set h3 = h2 h1 h2^{-1}. Check using Definition 2.6 (independence via gE(h)g^{-1} ≠ E(h')) that h3 is independent of both h1 and h2, and that h3 is strongly contracting WPD by conjugation invariance. If this holds, insert a one-sentence justification at the start of Section 4. If it fails in this generality, find the standard Schottky construction (e.g., large powers of h1 and h2) in cited works [Yan19] or [DGO17] that yields three pairwise independent WPD elements and add the citation. Either way, confirm that the constants in Lemma 2.33 and Section 4 can be chosen for the resulting three elements.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central counting machinery is built on a set F of three pairwise independent strongly contracting WPD elements. Section 4 begins: 'Let F ⊂ G be a set consisting of three pairwise independent strongly contracting WPD elements.' The Extension Lemma 2.33, the insertion map in Definition 4.4, and the multiplicity bound in Lemma 4.9 all require this three-element set. However, Theorem A assumes only two independent strongly contracting WPD elements. The paper never proves, cites, or remarks that from two such elements one can obtain three pairwise independent ones. Without this passage, the proof of Theorem A does not apply to its stated hypothesis: the growth-gap of short displacement (Theorem 4.1) and the subsequent genericity argument assume the three-element F. The gap is likely repairable: if h1,h2 are independent strongly contracting WPD, then h3 = h2 h1 h2^{-1} is again strongly contracting WPD, and independence of h3 from h1 and h2 follows from the definition of independence (if h3 were dependent with h1, then h2 would coarsely preserve the axis of h1, contradicting h2∉E(h1); similarly for h2). But as written, this step is omitted, so the proof of the theorem from the stated assumptions is incomplete. This is a load-bearing gap, not merely a cosmetic one, because the entire projection-complex construction and insertion argument hinge on F having three elements.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims three main results. First, Theorem A: if a finitely generated group admits an action by isometries on a geodesic metric space with two independent strongly contracting WPD elements, then strongly contracting WPD elements are exponentially generic in word-metric balls for every finite generating set. Second, Theorem B: generic elements have almost maximal stable word length and stable length in the auxiliary space. Third, Theorems D and E: a growth gap for short displacement holds, and finitely generated acylindrically hyperbolic groups are growth tight; Corollary 1.5 adds cogrowth tightness. The proof strategy is to build a projection complex on the axes of a finite set F of WPD elements, prove an anchored length estimate for word geodesics (Proposition 3.4), use an insertion-counting argument to get a growth gap for short projection-complex displacement (Theorem 4.1), and then use a guard-decomposition dichotomy to show that non-WPD elements are either short-displacement or conjugate to a shorter word (Lemma 5.9). Section 7 adapts an insertion argument from the companion paper [DY24] to prove uniform growth tightness for confined subgroups.","tokens_in":46931,"tokens_out":8586,"duration_ms":86371,"significance":"If the proof is completed, this is a substantial advance: exponential genericity of WPD elements in word-metric balls had been open for arbitrary finite generating sets in important settings such as mapping class groups and Out(F_n), and the anchored length estimate is a genuine counting analogue of linear progress. The main technical body, Sections 3–6, is detailed, carefully structured, and largely internally consistent: Proposition 3.4, Lemma 4.9, Lemma 5.9, and the stable-length arguments in Section 6 are nontrivial and appear to check out. The paper makes good use of existing projection-complex machinery (BBF15, BBFS19) and WPD divergence. However, the proof of the central Theorem A has a load-bearing gap in its stated hypotheses, and the growth-tightness section depends on companion-paper results in ways that are not fully verified in the text. These issues are likely repairable, but they must be fixed before the paper can be accepted.","major_comments":[{"comment":"Theorem A assumes only two independent strongly contracting WPD elements, but the proof of Theorem A begins in Section 4 by fixing 'a set F ⊂ G consisting of three pairwise independent strongly contracting WPD elements.' The Extension Lemma 2.33, the insertion map in Definition 4.4, and the multiplicity bound in Lemma 4.9 all require a three-element F. The paper never justifies the passage from two to three. This is not a cosmetic gap: the natural candidate h2 h1 h2^{-1} is conjugate to h1 and is therefore not independent of h1. A correct construction, for example using sufficiently high powers of h1 h2 to produce a third element independent of both, must be stated and proved. As written, the proof of Theorem A does not apply to the stated hypothesis.","section":"Section 4 and Theorem A"},{"comment":"The proof of growth tightness for acylindrically hyperbolic groups is not self-contained in the present manuscript. Lemma 7.12(ii) is only sketched and refers to [DY24, Lemma 4.8] for the complete argument; Lemma 7.13 is explicitly described as 'an abridged version of [DY24, Lemma 4.10]'; and Corollary 1.5 relies on Theorem 7.4 cited from the companion paper [DY26]. Since Theorem E and Corollary 1.5 are advertised applications, the referee cannot verify the key growth-tightness and cogrowth-tightness claims from the text alone. The authors should either include the missing arguments or clearly state that these results are conditional on the companion papers.","section":"Section 7, Theorem 7.10 and Corollary 1.5"},{"comment":"The proof of Lemma 5.8, which produces a bi-infinite standard path for a loxodromic element in P_K(F), relies on Lemma 5.7(2). The statement and proof of Lemma 5.7(2) are quite compressed and the notation u∈F_K(x,y), v∈F_K(y,z) is confusing; the proof of the concatenation property 'F_K[x,z]=F_K[x,y]·F_K[y,z]' appears to need a more careful verification. Since Theorem A uses Lemma 5.8 to identify WPD elements, this step is load-bearing. A fuller proof or a precise reference is needed.","section":"Section 5, Lemma 5.8"}],"minor_comments":[{"comment":"The letter o is used both for the basepoint in X and for the base vertex (the axis Ax(f0)) in P_K(F). The proof of Theorem D passes between the two via the map Φ without explicitly managing this overload; the intended identification should be flagged.","section":"Notation throughout"},{"comment":"The displayed inequalities contain unnamed constants C and c; in particular the C in Corollary 1.1 is not used in the displayed bound. The statements should specify that the error term is C λ^n.","section":"Corollaries 1.1 and 1.2"},{"comment":"The subsection heading 'Defining the insertion map' refers to '§4.4', but the insertion map for the projection complex is introduced in Section 4.1 of the present paper; this cross-reference is inaccurate.","section":"Section 7 text"},{"comment":"The paper cites [DY24] and [DY26] as preprints. If the companion papers are not yet published, the dependence should be made explicit in a footnote or introduction; otherwise the reader cannot evaluate the status of the cited lemmas.","section":"References"},{"comment":"Several displayed arrows are corrupted (e.g., the definition of the orbital map in Section 2.1), making the text difficult to read. This is presumably a typesetting issue but should be corrected.","section":"Section 2.1 and elsewhere"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely substantially correct in its core Sections 3–6, but the two-to-three WPD gap in the proof of Theorem A is a real defect in the central argument, and Section 7 delegates key steps to companion papers. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. The editor may wish to ask the authors to state the status of [DY24]/[DY26] clearly and to supply the missing two-to-three construction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Give this one to a good referee, but expect a revision. The main results — exponential genericity of strongly contracting WPD elements for every finite generating set, the stable-length large deviation, and growth tightness for acylindrically hyperbolic groups — are a real step forward. The anchored-length estimate and the positive-density WPD-axis recurrence (Theorem C) are genuinely new and clever; they replace weak-contraction assumptions used in earlier work and apply to arbitrary generating sets. The counting in Sections 3–6 is detailed and the algebra is consistent.\n\nThe soft spot is exactly what the stress-test flags: Theorem A assumes two independent strongly contracting WPD elements, but Section 4 starts with a set F of three pairwise independent such elements, and the extension lemma (2.33) and the insertion map need all three. The paper never explains how to obtain three from two. The specific fix suggested in the stress-test (conjugating one of the two) is wrong — that element is dependent with the original. The standard repair is to take h1^N h2^N for large N, which is strongly contracting WPD and independent of both; the authors should either state that or change the hypothesis of Theorem A to three elements. As written, the proof of Theorem A from the stated assumptions is incomplete, though I believe the gap is repairable.\n\nTwo smaller issues. Section 7 is heavily delegated to the companion papers [DY24, DY26]; Lemma 7.12(ii) and 7.13 are explicitly abridged. That makes Theorem E and Corollary 1.5 conditional on the correctness and availability of those papers. Also, Lemma 6.3 contains garbled unicode in the definition of A and D — a referee shouldn't have to guess the statement.\n\nThe citation pattern is fine; the self-citations point to genuinely used tools. Overall, this is important, serious work with a repairable but real gap at the base of the main theorem. Send it to a referee who knows projection complexes. If the two-to-three passage is fixed and the companion-paper dependencies are sorted, this should be published.","headline":"Strong counting paper with a real two-to-three WPD gap; the genericity theorem is likely true but the proof as written assumes what it needs to prove.","tokens_in":47339,"tokens_out":4373,"would_cite":true,"duration_ms":44640,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F67","37D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that in any finitely generated acylindrically hyperbolic group, WPD elements are exponentially generic with respect to every finite generating set, and derives growth and cogrowth tightness.","keywords":["acylindrically hyperbolic groups","WPD elements","exponential genericity","growth gaps","growth tightness","cogrowth tightness","stable translation length","projection complex"],"falsifier":"Examine the first paragraph of Section 4 and the remark following Lemma 2.33: the proof uses a three-element set F, while Theorem A assumes only two. To settle whether the result is true as stated, either construct a third independent WPD element from two, or find an action on a geodesic metric space with exactly two independent strongly contracting WPD elements and test whether the proportion of WPD elements in balls still tends to 1 exponentially.","tokens_in":46449,"feed_emoji":"📈","tokens_out":7377,"duration_ms":64297,"temperature":0.7,"pith_summary":"The paper proves a word-metric counting law for acylindrically hyperbolic groups — a broad class of groups with negative curvature — that was previously known only for random walks. For every finite symmetric generating set, elements satisfying the weak proper discontinuity condition (WPD elements) fill the balls of the Cayley graph with probability tending to 1 exponentially fast; equivalently, non-WPD elements have a strictly smaller exponential growth rate. The mechanism is a counting analogue of linear progress: generic word geodesics pass coarsely through linearly many WPD axes, ordered as they appear in a projection complex. From this the paper derives growth tightness and cogrowth tightness for the entire class, and exponential genericity of pseudo-Anosov elements in mapping class groups and fully irreducible elements in Out(F_n).","feed_headline":"WPD elements fill almost every word-metric ball","feed_subtitle":"In acylindrically hyperbolic groups, non-WPD elements have strictly smaller exponential growth, giving growth and cogrowth tightness.","key_machinery":"The central object is the projection complex P_K(F), a quasi-tree whose vertices are the G-invariant translates of finitely many strongly contracting WPD axes. Standard paths in this complex order the axes that any long word geodesic must coarsely visit. The argument is carried by three pieces: an anchored length estimate (standard-path length through chosen anchor points is at most the unanchored length plus a small error linear in the number of anchors), an insertion map that splices long WPD pieces into geodesic words at anchor positions with controlled multiplicity, and a guard decomposition of long word geodesics whose good blocks force coarse returns to WPD axes. Together they show tha","core_discovery":"The central claim is Theorem A: if a finitely generated group acts by isometries on a geodesic metric space and the action admits two independent strongly contracting WPD elements, then for every finite symmetric generating set S, μ_n({strongly contracting WPD elements}) ≥ 1 − C λ^n for some C>0 and 0<λ<1. The proof in fact establishes a stable-length large deviation (Theorem B): generic elements have stable length at least (1−ε) times their word-length displacement, both in the word metric and in the auxiliary space. The supporting dichotomy is that a generic element either has long displacement in the projection complex and is therefore WPD, or admits a conjugacy shortening; the latter set","pith_inferences":["The counting analogue of linear progress developed here should transfer to other counting problems in acylindrically hyperbolic groups, such as conjugacy growth or statistics of translation lengths, where random-walk results are already known.","The proof's three-element assumption suggests a concrete test: if two independent strongly contracting WPD elements can always be promoted to three pairwise independent ones, the theorem follows as written; if not, either a different argument is needed or the hypothesis of Theorem A must be strengthened.","The uniform growth gap for confined subgroups with a fixed confining set may extend beyond normal subgroups, giving control of Schreier graph growth for non-normal subgroups with the same confining data.","A natural next step is to make the constants C and λ explicit or to show they can be chosen uniformly over a family of generating sets; the current proof establishes existence only."],"forward_implications":["For every finite symmetric generating set of an acylindrically hyperbolic group, WPD elements are exponentially generic; consequently Morse elements (elements with coarsely geodesic cyclic subgroups) are exponentially generic as well.","Generic elements have almost maximal stable length: τ_S(g) ≥ (1−ε)|g|_S and τ_X(g) ≥ (1−ε)d(o,go), so the average stable word length over spheres tends to 1.","Every finitely generated acylindrically hyperbolic group is growth tight: for every infinite normal subgroup H, the quotient growth rate ω(G/H, S̄) is strictly smaller than ω(G,S).","Combined with the growth–cogrowth inequality of the companion paper, every infinite normal subgroup satisfies ω(H,S) > ½ ω(G,S), i.e. cogrowth tightness.","In mapping class groups and Out(F_n), pseudo-Anosov and fully irreducible elements are exponentially generic for every finite generating set, resolving previously open questions."],"fun_headline_variants":["Almost all elements are WPD in acylindrically hyperbolic groups","Non-WPD elements: exponentially small share","WPD genericity drives growth tightness","Exponential genericity of WPD elements"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that two independent strongly contracting WPD elements can be used where the proof fixes a set of three pairwise independent ones; Section 4 begins with 'Let F ⊂ G be a set consisting of three pairwise independent strongly contracting WPD elements' and the remark after Lemma 2.33 notes that the extension lemma requests three, but no argument supplies the third from the two assumed in Theorem A.","fun_headline_variants_meta":{"raw":{"variants":["Almost all elements are WPD in acylindrically hyperbolic groups","Non-WPD elements: exponentially small share","WPD genericity drives growth tightness","Exponential genericity of WPD elements"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000723,"raw_usage":{"total_tokens":2994,"prompt_tokens":570,"completion_tokens":2424,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":314,"completion_tokens_details":{"reasoning_tokens":2365}},"tokens_in":314,"tokens_out":2424,"duration_ms":19761,"temperature":1.0,"reasoning_tokens":2365,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T02:01:01.357193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Examine the first paragraph of Section 4 and the remark following Lemma 2.33: the proof uses a three-element set F, while Theorem A assumes only two. To settle whether the result is true as stated, either construct a third independent WPD element from two, or find an action on a geodesic metric space with exactly two independent strongly contracting WPD elements and test whether the proportion of WPD elements in balls still tends to 1 exponentially.","supporting_citations":[],"review_version":1}