Sublinear projection tracking in acylindrically hyperbolic groups

Lihuang Ding; Wenyuan Yang

Sublinear projection tracking in acylindrically hyperbolic groups

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper proves that, along any WPD isometry with a sublinearly contracting orbit, the shortest projection to the cyclic subgroup in the word metric and the pulled-back projection from the ambient space agree up to a sublinear error, and i

desk verdict A detailed, likely correct upgrade of projection tracking to WPD directions; the one thing to press is whether Sisto's Lemma 4.8 applies under the finite-WPD hypothesis. read the letter →

arxiv 2607.25555 v1 pith:PKQBYHQ5 submitted 2026-07-28 math.GR math.GT

Lihuang Ding , Wenyuan Yang This is my paper · ORCID

classification math.GRmath.GT MSC 20F6520F6737D40

keywords acylindricallyhyperbolicgroupsMorseelementsWPDisometriessublinearcontractionprojectiontrackinggrowthfunctionsgrowth–cogrowthinequalityquotientsof

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

This paper aims to show that acylindrically hyperbolic groups, even when they admit no distance formula, still have a robust projection structure along Morse directions. The central claim is that for a WPD isometry h whose orbit is strongly sublinearly contracting, the shortest projection to the cyclic subgroup ⟨h⟩ in the word metric tracks, up to sublinear error, the pullback of shortest projection to the axis in the auxiliary metric space. That tracking property is strong enough to power counting arguments: it gives exponentially sharp upper bounds on subgroup and group growth, constructs proper quotients whose growth rates converge to the original group's growth rate, and proves a growth–cogrowth inequality for confined subgroups. A reader should care because these results were previously known only under strong contraction or bounded tracking error, which general acylindrically hyperbolic groups need not satisfy.

What carries the argument

The central objects are the two projections: the shortest projection Pr_S^H in the Cayley graph and the pullback Pr_X^H of shortest projection to the axis Ho in X. The paper proves they differ by a sublinear error. The engine of the proof is a generalized Morse lemma: for every Lipschitz path with the same endpoints as a Morse geodesic γ, and every positive-density subinterval p of γ, the path intersects the r-neighborhoods of a sequence of pairwise R-separated points on p, with the number of such points linear in ℓ(p)/R. Together with a geometric separation lemma from the WPD literature, this forces word geodesics to spend long stretches near the axis Ho whenever their X-projections do, whi

What would settle it

A concrete way to falsify the theorem would be to exhibit a finitely generated group acting on a metric space where a WPD isometry has a strongly sublinearly contracting orbit, together with elements g_n for which the word-metric projection to ⟨h⟩ and the pulled-back axis projection differ by at least ε·d_S(g_n,⟨h⟩) for a fixed ε>0. The proof forces such a configuration to violate the geometric separation lemma — two word-close points whose orbits sit near widely separated axis points while remaining far from the axis — so any example with that configuration would settle the claim.

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Extended reading notes

Core claim

The load-bearing result, Theorem A, states: let G be a finitely generated group acting by isometries on a metric space X, and let h ∈ G be a WPD isometry (a weak properness condition along the orbit) such that the orbit ⟨h⟩o is strongly κ-contracting in X for some sublinear function κ. Then (1) the action has the sublinear projection tracking property along ⟨h⟩, meaning diam_S(Pr_S^H(g) ∪ Pr_X^H(g)) ≤ κ_track(d_S(g,H)) for all g; and (2) the pulled-back projection Pr_X^H is strongly sublinearly contracting in the word metric, meaning projections of two points depend on their word distance rather than on the distance to H. The proof combines a generalized Morse lemma, showing that any Lipschi

Load-bearing premise

The proof's load-bearing premise is an external geometric separation lemma, quoted without proof, that whenever two group elements have orbit points near widely separated points of the axis yet are close in the word metric, they must be close to the subgroup; if that lemma needs extra hypotheses such as geodesicity of the ambient space, or its constants fail to line up with the sublinear estimates, the central theorem does not follow.

Editorial extensions

If this is right

For any non-virtually cyclic subgroup K containing a generalized loxodromic element, the growth function of K satisfies |B_n ∩ K| ≤ e^{nω(K,S)+κ(n)}; in particular ω(K,S) is a genuine limit, not merely a limsup.
For acylindrically hyperbolic groups, there exist proper quotients with infinite kernels whose growth rates converge up to the growth rate of the original group, showing no uniform gap between a group and its quotients.
For mapping class groups of closed surfaces of genus at least 2, the ball count is bounded by n^d e^{nω}, a polynomial correction on top of purely exponential growth.
The growth–cogrowth inequality holds for confined subgroups in both acylindrically hyperbolic groups and Morse local-to-global groups with Morse elements; as a corollary it holds for every infinite normal subgroup, and whenever the Schreier graph has subexponential growth, the subgroup has full exponential growth.

Reading between the lines

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

Because the proof of Theorem A uses acylindricity along the subgroup and the Morse property of the axis rather than hyperbolicity of X, a natural test is whether Lemma 4.10's 'strongly sublinearly contracting' assumption can be weakened to ordinary sublinear contraction; the paper states this is not known.
The combination of the no-uniform-gap theorem with the growth–cogrowth inequality suggests that any eventual proof of growth tightness for acylindrically hyperbolic groups cannot rely on a universal positive gap, and may need to track the sublinear error function rather than a constant.
The paper's polynomial correction for mapping class groups is consistent with a general principle that logarithmic projection tracking yields polynomial-times-exponential growth; under that heuristic, constructing a geodesic automatic combing would pin down whether the polynomial factor is a real geometric invariant or an artifact of the proof.

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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

First thing to know: this is a substantial paper, and if Theorem A holds up, it is a real step forward. It replaces bounded contraction with sublinear tracking along WPD directions, and the applications to growth, quotients, and cogrowth are concrete. The proofs are written out in full, not sketched, and there is no visible circularity.

What's genuinely new: Theorem A's sublinear projection tracking for WPD elements is not in the cited literature; the generalized Morse lemma (Lemma 4.4) is a real strengthening of middle recurrence; and the effective growth bounds and quotient approximations are new. The extension to Morse local-to-global groups in Theorem D is a nice addition.

Soft spots, in proportion: The load-bearing external input is Lemma 4.8, quoted from Sisto [Sis16, Lemma 3.4]. It is not reproduced, and the paper's Definition 4.6 only requires the WPD intersection to be finite, not uniformly bounded. If Sisto's lemma in fact needs a uniform acylindricity bound, then the proof of Theorem A as stated for arbitrary metric spaces has a gap. Remark 4.7 claims the hyperbolic proof extends to general metric spaces using only Morse-ness, but the details are not given. For the main applications—acylindrically hyperbolic groups acting on hyperbolic spaces—the uniform version should hold, so Theorems B–D are likely safe even if Theorem A needs a hypothesis tweak. A referee should verify this matching explicitly. This is the one issue worth pressing; nothing else in the paper looks broken.

The paper is long, and the estimates are complex enough that independent verification is hard—the reader's moderate confidence is fair. There are also a few LaTeX artifacts (garbled arrows in remarks) but nothing substantive.

Who it's for: people working on counting, growth, and projections in acylindrically hyperbolic groups. It deserves a serious referee. I'd send it to review with the instruction to check the Sisto lemma application and, if needed, ask the authors to either reproduce the needed formulation or restrict Theorem A to the uniform acylindricity setting.

Referee Report

0 major / 5 minor

Summary. The paper establishes a 'sublinear projection tracking' theorem for finitely generated groups acting on metric spaces with a WPD isometry whose orbit is strongly sublinearly contracting (Theorem A). The two main ingredients are a generalized Morse lemma for contracting geodesics and a geometric separation lemma of Sisto; together they show that the word-metric shortest projection to the cyclic subgroup tracks the pullback of the shortest projection in the auxiliary space, and that the pulled-back projection is strongly sublinearly contracting in the word metric. The applications are: effective upper bounds on relative growth functions of acylindrically hyperbolic groups (Theorem B), construction of proper quotients whose growth rates converge to the ambient growth rate (Theorem C), and a growth--cogrowth inequality for confined subgroups in acylindrically hyperbolic groups and in Morse local-to-global groups with Morse elements (Theorem D).

Significance. If correct, this is a substantial contribution. Acylindrically hyperbolic groups do not in general admit a Masur--Minsky-type distance formula, and the paper supplies a robust, quantitative substitute along WPD directions that is strong enough to drive counting and quotient constructions. The growth estimates and the growth--cogrowth inequality extend previously known results from the strongly-contracting setting to the broader WPD/Morse setting. I read the paper in good faith and found no circularity: Theorem A is proved independently and the applications genuinely use it as an input. The proof is unusually complete for a paper of this scope: the generalized Morse lemma, the two projection-tracking lemmas, the free-semigroup construction, the quotient argument, and the Poincar\'e-series estimate in Section 7 are all written out in detail. The external reliance on Sisto's Lemma 4.8 is explicit and legitimate: Definition 4.6 matches the acylindricity-along-H hypothesis used in [Sis16, Lemma 3.4], and the additional properness of \pi|_H follows from the quasi-geodesic orbit assumption in Theorem A. The skeptic's concern about this dependency does not, on reading the paper, land as a

minor comments (5)

[§4.1, Eq. (4.3)] In the proof of Lemma 4.4, Lemma 2.12 as stated gives a bound involving both \ell(\beta_i) and d(\beta_i,p), but Eq. (4.3) suppresses the distance term. Please add the short justification that d(\beta_i,p) is comparable to \ell(\beta_i) for the components under consideration and that the constant \theta_0 is chosen small enough to absorb it. This is a proof-reading clarification rather than a substantive obstruction.
[§2.1 and Lemma 4.10] The displayed definition of d^Y_A appears to have a typo: the second formula should use diam_Y, not diam_X. Also, in the proof of Lemma 4.10 the phrase 'd^X_H(g,h)=d_S(x,y)' conflates the X-diameter with the word distance; the intended statement is that d_S(Pr^X_H(g), Pr^X_H(h)) is controlled via the quasi-isometric embedding \pi|_H. Please rephrase.
[Lemma 1.1 vs Lemma 4.1] The introductory generalized Morse lemma states N=\lceil \ell(p)/6R\rceil, while the proved Lemma 4.1 (and its use in Lemmas 4.9--4.10) gives N=\lfloor \ell(p)/6R\rfloor. The difference is harmless, but the statements should be aligned.
[§5.1, Lemma 5.4] The proof refers to '(5.1)' at a point where no equation (5.1) has yet been defined. The intended reference appears to be the inequality just proved or the later Proposition 5.2 estimate; please correct the numbering.
[Throughout] There are many corrupted LaTeX arrows and symbols, e.g. '/leftr⫯g⊸tl⫯ne→' in Remark 1.6, Lemma 2.11, Lemma 3.7, §5, and §7.2--7.3, and similar '/leftfootl⫯ne→' artifacts. These should be replaced by proper arrows before publication.

Circularity Check

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No circular derivation: Theorem A is proved from the WPD hypothesis via Sisto's external separation lemma and a new Morse lemma; applications invoke it after proof.

full rationale

I walked the derivation chain from Theorem A through Lemmas 4.9 and 4.10 to the applications in Theorems B, C, and D. Theorem A is not assumed in its own proof: Lemma 4.9 proves sublinear projection tracking from the WPD/acylindrical-along-H hypothesis, Sisto's Lemma 4.8, and the paper's own Lemma 4.4; Lemma 4.10 proves strong sublinear contraction of the pulled-back projection from the same external separation result and the strong contraction assumption. The applications then use Theorem A as an already-proved input, so there is no fitted parameter renamed as a prediction and no quantity defined in terms of the claimed conclusion. The paper does cite prior work by Yang and coauthors—for example [Yan22, Lemma 8.8] in the quotient construction and [CGY24] in the growth–cogrowth section—but these are published, independent tools used as inputs, not the target theorem itself, and they do not force the main result by construction. The heaviest external dependency, Sisto's Lemma 4.8, is quoted rather than reproved; this is a correctness/verification risk if the hypotheses do not match, but it is not circularity because the lemma is an external result, not a restatement of the paper's conclusion. No step was found where an equation reduces to an input by definition or where a self-citation substitutes for the central argument.

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

No empirical fitted parameters or invented entities are introduced. The constants and sublinear functions appearing in the proof are existential proof variables, not numbers fitted to data. The external assumptions are standard results or theorem hypotheses.

assumptions (4)

domain assumption Sisto's geometric separation lemma ([Sis16, Lemma 3.4], quoted as Lemma 4.8): under acylindricity along H and properness of π|_H, for every r and D there is M such that elements whose orbit points lie near separated points in π(H) and which are D-close in the word metric are M-close to H.
Unproved in this paper; the main technical Lemmas 4.9 and 4.10 rely on it. It is an external published result, not the target result.
domain assumption Morse quasi-geodesics are characterized by middle recurrence and sublinear contraction (Lemma 2.11, citing [ACGH17], [DMS10], [ADT17]).
Used throughout to convert Morse properties into projection bounds; external to this paper.
domain assumption WPD condition / acylindricity along H and strong κ-contraction of the orbit are hypotheses of Theorem A.
These are stated as assumptions of the main theorem, not derived; they are standard in the field.
domain assumption Morse local-global assumption [RST22, Definition 2.12] and existence of Morse elements for type II groups.
Used in Section 7 for the growth–cogrowth inequality in Morse local-to-global groups.

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Pith. "Pith review of Sublinear projection tracking in acylindrically hyperbolic groups." pith.science (2026). https://pith.science/paper/PKQBYHQ5

@misc{pith2026260725555,
  author       = {Pith},
  title        = {Pith review of: Sublinear projection tracking in acylindrically hyperbolic groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PKQBYHQ5}},
  note         = {Machine review of arXiv:2607.25555}
}

read the original abstract

We study projection phenomena in word metrics of finitely generated acylindrically hyperbolic groups. For a loxodromic WPD element acting on a hyperbolic space, we prove that shortest projection in the word metric to the corresponding cyclic subgroup sublinearly tracks the pullback of shortest projection to its axis in the hyperbolic space. As applications, we obtain effective upper bounds for growth functions and construct proper quotients whose growth rates converge to that of the original group. We further prove a growth--cogrowth inequality for confined subgroups in both acylindrically hyperbolic groups and Morse local-to-global groups with Morse elements.

Figures

Figures reproduced from arXiv: 2607.25555 by the authors.

**Figure 1.** Illustration of κ-contraction (left) and strong κ-contraction (right). and many groups acting on CAT(0) spaces. For general acylindrically hyperbolic groups, no comparable distance formula is available. The auxiliary hyperbolic space associated with an acylindrical action may be very far from the Cayley graph, and the orbit map usually loses too much information to recover the word metric additively. The main point… view at source ↗

**Figure 2.** Lemma 4.4: decompose the path α from x to y into components αi contained in Nr/2(p) (the grayed region). Let βi = [uio, xio]X ⋅ π(αi) ⋅ [yio, vio]X. This path joins uio to vio and has length at most 3λNi . Thus, for c = 3λ/ϵ, ℓ(βi) ≤ c dX(uio, vio), so βi is a c-Lipschitz path. Recall that γ = Ho is θ-middle recurrent for some θ > 0. Thus the θ-middle of [uio, vio]γ meets the R-neighborhood of βi for a constant R = … view at source ↗

**Figure 3.** Lemma 4.9 It suffices to prove that dX(o, yo) ∥x −1y∥S Ð→ 0 as ∥x −1 y∥S → ∞. Indeed, if n ∶= ∥x −1 y∥S and dX(o, yo) = o(n), then ∥y∥S = o(n), while ∥x∥S = dS(x, H) ≤ n ≤ ∥x∥S +∥y∥S. Thus n ∼ dS(x, H) and the required tracking estimate follows. We argue by contradiction. Let l ∶= dX(o, yo). There exists ϵ > 0 so that l ≥ ϵn holds for infinitely many pairs (x, y) with n → ∞. Consider the path α ∶= π([1, x]S) labeled… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Lemma 4.10 Recall that N ≥ ℓ(p)/6R−1 ≥ ϵn/12R−1 ≥ ϵn/24R when n ≫ 1. Choose D satisfying Lemma 4.8 large enough so that ϵD 24R > 1. This gives a contradiction: dS(1, x) ≥ ϵD 24R ⋅ n > n. The proof is complete. □ We now establish that, under a stronger assumption, the p…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Largest acylindrical actions and stability in hierarchically hyperbolic groups

Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham. Largest acylindrical actions and stability in hierarchically hyperbolic groups. Trans. Amer. Math. Soc. Ser. B , 8:66--104, 2021. With an appendix by Daniel Berlyne and Jacob Russell

2021

[2] [2]

Arzhantseva, C

G. Arzhantseva, C. Cashen, D. Gruber, and D. Hume. Characterizations of M orse quasi-geodesics via superlinear divergence and sublinear contraction. Doc. Math. , 22:1193--1224, 2017

2017

[3] [3]

Tarik Aougab, Matthew Gentry Durham, and Samuel J. Taylor. Pulling back stability with applications to Out (F_n) and relatively hyperbolic groups. J. Lond. Math. Soc. (2) , 96(3):565--583, 2017

2017

[4] [4]

Abbott and Jason F

Carolyn R. Abbott and Jason F. Manning. Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups. Michigan Math. J. , 74(2):357--402, 2024

2024

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Connections between the topology of the morse boundary, the morse local-to-global property and acylindrical hyperbolicity, 2026

Carolyn Abbott and Stefanie Zbinden. Connections between the topology of the morse boundary, the morse local-to-global property and acylindrical hyperbolicity, 2026

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Balasubramanya

S. Balasubramanya. Acylindrical group actions on quasi-trees. Algebr. Geom. Topol. , 17:2145--2176, 2017

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Bestvina and K

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Hierarchically hyperbolic spaces II : Combination theorems and the distance formula

Jason Behrstock, Mark Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces II : Combination theorems and the distance formula. Pacific Journal of Mathematics , 299(2):257--338, 2019

2019