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Growth gaps and exponential genericity in acylindrically hyperbolic groups

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.25578 v1 pith:4Z6HXIEG submitted 2026-07-28 math.GR math.GT

Lihuang Ding , Wenyuan Yang This is my paper · ORCID
classification math.GRmath.GT MSC 20F6520F6737D40
keywords acylindricallyhyperbolicgroupsWPDelementsexponentialgenericitygrowthgapstightnesscogrowthstabletranslationlengthprojectioncomplex
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

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).

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 5 minor

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.

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 (3)
  1. [Section 4 and Theorem A] 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.
  2. [Section 7, Theorem 7.10 and Corollary 1.5] 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.
  3. [Section 5, Lemma 5.8] 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.
minor comments (5)
  1. [Notation throughout] 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.
  2. [Corollaries 1.1 and 1.2] 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.
  3. [Section 7 text] 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.
  4. [References] 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.
  5. [Section 2.1 and elsewhere] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction in the main counting/genericity argument; the central proof is self-contained apart from a two-to-three WPD-element gap that is an incompleteness, not circularity. Some application-level dependence on companion papers is present but does not make the derivation circular.

full rationale

The central derivation of Theorem A does not assume its own conclusion. The counting machinery (anchored length, insertion map, kernel estimate, guard decomposition) is built from external tools (BBF15, BBFS19, Yan14, Sis16, GS23) and from the paper's own projection-complex construction; the genericity conclusion is not used as an input. The only potentially circular-looking steps are self-citations, but they are not circular under the stated rules: (i) Theorem E's growth tightness is proved by adapting the insertion argument of [DY24] and using the confined extension lemma from [CGY24]/[DY26]; these are cited theorems with stated hypotheses not containing the target result, so they count as independent support even though authors overlap. (ii) Corollary 1.5 is literally obtained by combining Theorem E with [DY26, Theorem D]: the text says 'In [DY26] we prove that ... We therefore obtain the following corollary by Theorem E.' This is an application-level self-citation, but it is a logical combination of external theorems, not a reduction of the paper's conclusion to itself. A genuine and load-bearing gap, unrelated to circularity, is the passage from two independent strongly contracting WPD elements in Theorem A to the set F of three pairwise independent such elements used in Section 4. The paper explicitly notes: 'We remark that Lemma 2.33 requests F to contain three independent elements,' and Section 4 begins 'Let F ⊂ G be a set consisting of three pairwise independent strongly contracting WPD elements.' Theorem A assumes only two, and the paper never proves or cites the two-to-three passage. This is an omitted justification/incomplete proof, not a circular one: the argument does not assume the theorem's conclusion, it works from a stronger hypothesis than stated. I therefore keep the circularity score low but flag this gap for verification.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on established projection-complex and contraction machinery; the only self-postulated ingredients are the proof constants and the companion-paper results.

free parameters (4)
  • K (projection complex constant) = K > max{4θ+β+5C, N+4θ} (Eq. 3.1)
    Chosen large enough for projection axioms, Lemma 2.37, and divergence thresholds; all later estimates depend on this choice.
  • ε (Theorem 4.1 displacement threshold) = 1/(4|S|^l M)
    Explicit value chosen so that the insertion-counting ratio has base < 4/5; depends on generating set and WPD stabilizer bound.
  • δ and m (exponential decay rate) = δ = min{1/100, ε/E}, m = δn
    Balancing parameters in the proof of Theorem 4.1 to force exponential decay; not parameters of the theorem statement.
  • R (bad-block radius) = R chosen so that κ(r)ε > 1 (Lemma 5.5)
    Chosen in Lemmas 5.5 and 6.4 to make bad guarded blocks rare; affects constants in Theorem B.
assumptions (6)
  • standard math Projection complex theory: P_K(F) is a quasi-tree, standard paths have bottleneck and almost-tripod properties, and G acts acylindrically on P_K(F) (BBF15, BBFS19).
    Foundational for Sections 3–7; used via Lemma 2.20, 2.21, Theorem 2.13, Lemmas 2.36 and 2.37.
  • standard math Admissible path fellow-travel: every (L,τ)-admissible path with long local pieces is a uniform quasi-geodesic with the fellow-travel property (Yan14, Proposition 2.10).
    Used in Extension Lemma 2.33, Lemma 4.5, and Lemma 7.12.
  • domain assumption Strongly contracting WPD elements are (κ,N)-divergent in the word metric (Lemma 2.32, via BBFS19/GS23).
    Critical for lower bounds on lengths of paths avoiding the axis in Lemmas 3.9, 5.5, and 6.4. The paper proves this lemma using the quasi-tree of spaces.
  • standard math Sisto's theorem: strongly contracting WPD elements are Morse in the word metric (Lemma 2.29(3)).
    Used to convert WPD genericity to Morse genericity and to define maximal elementary subgroups E(f).
  • domain assumption Insertion map injectivity/imaging lemmas from companion paper [DY24] (Lemmas 4.4, 4.8, 4.10).
    The proof of Theorem 7.10 (growth tightness) abridges these; the paper refers to [DY24] for complete proofs.
  • domain assumption Growth-cogrowth inequality ω(H,S)+1/2ω(G/H,S̄) ≥ ω(G,S) from companion paper [DY26, Theorem D].
    Needed to derive Corollary 1.5 (cogrowth tightness); not proved in this paper.

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Pith. "Pith review of Growth gaps and exponential genericity in acylindrically hyperbolic groups." pith.science (2026). https://pith.science/paper/4Z6HXIEG

@misc{pith2026260725578,
  author       = {Pith},
  title        = {Pith review of: Growth gaps and exponential genericity in acylindrically hyperbolic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Z6HXIEG}},
  note         = {Machine review of arXiv:2607.25578}
}
read the original abstract

We prove that, for every finite generating set of an acylindrically hyperbolic group, the set of non-WPD elements has strictly smaller exponential growth rate. Equivalently, WPD elements are exponentially generic. As applications, we prove growth tightness and cogrowth tightness for acylindrically hyperbolic groups.

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