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Hierarchically hyperbolic groups and uniform exponential growth

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Pith's one-line read Virtually torsion-free hierarchically hyperbolic groups either have uniform exponential growth or are quasi-isometric to $\mathbb{Z}\times E$; if the dichotomy holds, acylindrically hyperbolic HHGs grow uniformly.

desk verdict A strong structural theorem for hierarchically hyperbolic groups with a repairable gap in Proposition 4.2 and a constants bookkeeping issue; worth serious refereeing. read the letter →

arxiv 1909.00439 v2 pith:T4A6DTH4 submitted 2019-09-01 math.GR math.GT

Carolyn Abbott , Thomas Ng , Davide Spriano , Radhika Gupta , Harry Petyt This is my paper · ORCID
classification math.GRmath.GT MSC 20F6520F6720F69
keywords hierarchicallyhyperbolicgroupsuniformexponentialgrowthacylindricallyCAT(0)cubicalquantitativeTitsalternativebigsettranslationlengthquasi-isometricproduct
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 dichotomy for virtually torsion-free hierarchically hyperbolic groups (HHGs), a broad class that includes hyperbolic groups, mapping class groups, and many CAT(0) cubical groups. Every such group either has uniform exponential growth—word counts grow exponentially at a rate bounded away from zero for every finite generating set—or its Cayley graph is quasi-isometric to $\mathbb{Z}\times E$ for some space $E$. If the dichotomy is right, every virtually torsion-free acylindrically hyperbolic HHG has uniform exponential growth, and certain CAT(0) cubical groups of dimension at least three receive their first proof of it. The paper also records a structural description of the exceptional non-uniform case and a quantitative Tits alternative under extra hypotheses.

What carries the argument

The proof runs on the hierarchical structure itself: the group is encoded by an index set of domains, each carrying a hyperbolic coordinate space $C_U$, with projection maps satisfying consistency axioms. The load-bearing objects are the big set $\operatorname{Big}(g)$, the set of domains on which a group element acts with unbounded orbit, and a uniform lower bound $\tau_0>0$ on the stable translation length of every infinite-order element on each domain in its big set (Lemma 2.23). With that bound, the ping-pong exponents and word-length bounds become independent of the generating set. The dichotomy is decided by whether two short words have big sets meeting non-orthogonal domains: if so, ping-pong on the projected hyperbolic spaces produces a uniformly short free semigroup; if not, the domains in the generating set's big sets form a $G$-invariant family of pairwise orthogonal domains, forcing a quasi-isometric product decomposition $\mathbb{Z}^{|B|}\times E$.

What would settle it

Exhibit a virtually torsion-free HHG and a sequence of infinite-order elements $g_n$ with a domain $U_n\in\operatorname{Big}(g_n)$ such that the stable translation length $\tau_{U_n}(g_n)$ tends to zero; that would directly refute Lemma 2.23. Alternatively, find a sequence of generating sets for one HHG with no two elements of length bounded by any fixed $M$ generating a free semigroup, while the Cayley graph is not quasi-isometric to $\mathbb{Z}\times E$.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: let $(G,S)$ be a virtually torsion-free hierarchically hyperbolic group. Then either $G$ has uniform exponential growth, or there is a space $E$ such that the Cayley graph of $G$ is quasi-isometric to $\mathbb{Z}\times E$. The proof in fact establishes a trichotomy: for every generating set either two words of uniformly bounded length generate a free semigroup, or $G$ is virtually abelian, or $G$ is quasi-isometric to $\mathbb{Z}^{|B|}\times E$ with $B$ a $G$-invariant collection of pairwise orthogonal domains whose associated hyperbolic spaces are uniformly quasi-lines. From this, Corollary 1.3 follows: virtually torsion-free HHGs that are acylindrically hyperbolic have uniform exponential growth. The authors also show that when the top-level hyperbolic space $C_S$ is non-elementary, the free semigroup can be upgraded to a genuine free subgroup, giving a quantitative Tits alternative.

Load-bearing premise

The proof assumes a fixed positive lower bound $\tau_0$ on the stable translation length of any infinite-order element on any domain in its big set, with $\tau_0$ independent of the generating set; if some element's translation length on a big domain can be arbitrarily small, the short-word constants collapse and uniformity is lost.

Editorial extensions

If this is right

  • Every virtually torsion-free HHG whose Cayley graph is not quasi-isometric to a nontrivial product has uniform exponential growth.
  • Virtually torsion-free acylindrically hyperbolic HHGs have uniform exponential growth; this covers non-elementary hyperbolic groups, non-exceptional mapping class groups, and many CAT(0) cubical groups.
  • An HHG with an asymptotic cone containing a cut-point, or with an unbounded Morse quasi-geodesic, has uniform exponential growth.
  • If the top-level hyperbolic space $C_S$ is non-elementary, and $G$ is not quasi-isometric to $\mathbb{Z}\times E$, then every generating set contains two elements of uniformly bounded length generating a free subgroup.
  • A virtually torsion-free HHG without uniform exponential growth has a very restricted shape: a $G$-invariant set of pairwise orthogonal domains whose hyperbolic spaces are uniformly quasi-lines, with all other unbounded domains orthogonal to them.

Reading between the lines

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

  • The dichotomy suggests that any counterexample to Gromov's question inside the HHG class would have to be virtually abelian or quasi-isometric to $\mathbb{Z}\times E$; ruling out exponential non-uniform growth among such products would close Question 1.9 affirmatively.
  • Because the constants depend only on the hierarchy constants and $\tau_0$, the theorem yields explicit, computable growth bounds for any HHG whose hierarchy data are known; one could turn the proof into an algorithm that, given a presentation and hierarchy constants, outputs the uniform word-length bound.
  • If all CAT(0) cubical groups are HHGs, as conjectured, the result would give uniform exponential growth for every virtually torsion-free cubical group that is not quasi-isometric to $\mathbb{Z}\times E$, extending the first-proof status beyond dimension three.
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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

2 major / 4 minor

Summary. The paper studies uniform exponential growth for virtually torsion-free hierarchically hyperbolic groups (HHGs). The main theorem (Theorem 1.1) states that every such group either has uniform exponential growth or its Cayley graph is quasi-isometric to a product of the form Z×E. The proof strategy is a dichotomy: if short words act loxodromically on two non-orthogonal domains, ping-pong arguments produce a uniformly short free subsemigroup; otherwise the paper shows that a certain set of domains is an invariant pairwise-orthogonal collection, and a structural analysis yields either uniform exponential growth, virtual abelianness, or a product decomposition. The paper derives several corollaries, including uniform exponential growth for acylindrically hyperbolic HHGs, for groups with Morse or quasi-convex subgroups, and a quantitative Tits alternative under hierarchical acylindricity.

Significance. If correct, the main theorem is a significant contribution: it provides a new unified proof of uniform exponential growth for several classes of non-positively curved groups, including the first proof for certain CAT(0) cubical groups of dimension at least three, and it gives a quasi-isometric restriction on any virtually torsion-free HHG that fails to have uniform exponential growth. The structural theorem for the non-uniform-growth case (Theorem 1.10) is a strong, falsifiable statement, and the use of consistent tuples to obtain product decompositions is a useful contribution. The paper also gives credit to the relevant background machinery and does not appear to be circular; however, one load-bearing self-citation, Lemma 2.23 from the authors' earlier paper [AB18], supplies the uniform translation-length lower bound that makes the ping-pong constants independent of the generating set.

major comments (2)
  1. [§4, Proposition 4.2] In the branch where B is not G-invariant, the proof asserts that |X^N.B| >= N+1 contradicts the fact that there are at most N pairwise orthogonal elements. This contradiction is only valid if X^N.B is pairwise orthogonal, but the proof does not establish this: B is a union of Big(s) over generators, and images of different domains under different words need not be orthogonal. The argument can be repaired by splitting into subcases: if X^N.B contains two non-orthogonal domains V1,V2, then conjugating the corresponding generators by words of length at most N gives elements of length at most 2N+1 with non-orthogonal big-set domains, which is exactly the desired conclusion; otherwise X^N.B is pairwise orthogonal and the cardinality contradiction applies. As written, the dichotomy is not proven. In addition, the statement of Item 1 says elements lie in X^N, while the proof actually produces elements in X^{2N+1}; the later text in Section 4.1 uses the latter bound.
  2. [§4, Proof of Theorem 4.1] The constant M is chosen as max{k1, 2n0+k2, k3+2, 3(k4+2)(N+1)!}, but this does not control the X-lengths of the elements that actually generate the free semigroup. In Case 1 the generators are powers of s and t with |s|,|t| <= 2N+1, so their X-lengths can be as large as (2N+1)k1, not merely k1. In the nested case, the word t^{n0} s^{k2} t^{-n0} has X-length at most 2N n0 + N k2, not 2n0 + k2. The proof of Proposition 4.4 also says to replace 2N+1 by 2n0+1, which ignores the factor N carried by s. The theorem would still hold after enlarging M by a factor depending only on N and the hierarchy constant, but as written the asserted uniform word-length bound in Theorem 4.1 does not follow from the displayed choice of M.
minor comments (4)
  1. [§4, Proposition 4.3] In the ping-pong estimate, the text writes d_V(ρ_U^V, t^{k(2N+1)!} x) >= τ0 |k|, but the exponent is k(2N+1)! and the lower bound should involve τ0 k(2N+1)!; the displayed estimate appears to omit the factor (2N+1)!.
  2. [§4, Proof of Proposition 4.4] The phrase 'replacing 2N+1 with 2n0+1' is inaccurate because s has X-length at most N; the correct replacement should involve max(N, 2n0+N).
  3. [Corollary 1.4] The proof of Corollary 1.4 asserts that if G is quasi-isometric to a product with unbounded factors, then an infinite quasi-convex subgroup is either coarsely dense in G or has bounded diameter. This dichotomy is not justified in the text; please add a proof or a reference.
  4. [Throughout] There are several minor typographical issues: 'By By Proposition 2.27' in the proof of Theorem 4.1, 'CAT(0) cubical groups' in the abstract where 'CAT(0) cubical spaces' is meant, and in Example 1.7 the statement that a group 'is isometric to the product of two trees' should presumably refer to its Cayley graph.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main dichotomy is proved from HHG axioms and external results, not from the conclusion.

full rationale

The central claim (Theorem 1.1 / Theorem 4.1) is a dichotomy: either uniform exponential growth or a quasi-isometric product decomposition with a G-invariant set of pairwise orthogonal domains. The proof does not define either branch in terms of the other, and no parameter is fitted to the target conclusion. Case 1 of Proposition 4.2 produces short free-semigroup generators via the ping-pong lemma and the Breuillard--Fujiwara criterion, with constants depending only on the hierarchy constant, the orthogonality bound N, and the uniform translation-length lower bound. Case 2 constructs the product Z^|B| × E through Proposition 2.27 using consistent tuples and the distance formula, after Proposition 3.3 shows the relevant curve graphs are quasi-lines; this is a structural construction rather than a renamed assumption. The only load-bearing self-citation is Lemma 2.23, quoted from [AB18, Lemma 1.8], which gives a uniform lower bound on translation lengths of infinite-order elements on their big domains. That lemma is independent external support: it is parameter-free for all hierarchically hyperbolic groups, its assumptions do not include uniform exponential growth, and it is not derived from Theorem 1.1. Under the review rules, such a citation does not raise the circularity score. A referee should separately note the correctness gap in Proposition 4.2's non-G-invariant branch, where the 'at most N pairwise orthogonal elements' bound is applied to X^N.B without establishing pairwise orthogonality, and where the statement promises X^N while the proof produces elements of length at most 2N+1. That is a rigor issue requiring a case split and adjusted constants, not a circularity, since repairing it would not identify the theorem's output with its input. No circular step was found.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It relies on a large body of prior HHG theory, including work by some of the authors (AB18, RST18, Spr17), but these are external theorems, not assumptions tailored to the target result.

assumptions (8)
  • domain assumption The HHG axioms (Definition 2.8), including bounded domain dichotomy and finite complexity
    The entire proof operates within this axiomatic framework.
  • standard math All HHGs can be normalized (DHS17 Proposition 1.16)
    Standing assumption in Section 2.3; used to ensure projections are coarsely surjective.
  • standard math Classification of hierarchical automorphisms: every element is elliptic or axial, and big sets are pairwise orthogonal (DHS17 Proposition 2.21)
    Used in Proposition 4.2 and throughout to define big sets and pass to powers fixing domains.
  • standard math Uniform lower bound on translation lengths (AB18 Lemma 2.23)
    Used to make ping-pong exponents independent of the generating set.
  • standard math Breuillard-Fujiwara Proposition 11.1 (free semigroups from independent loxodromics)
    Used to produce free semigroups in Theorem 4.1.
  • standard math Fujiwara Proposition 2.3(2) (free subgroups from acylindrical actions)
    Used for the quantitative Tits alternative, Proposition 1.8.
  • standard math The action of an HHG on its top-level hyperbolic space is cobounded and acylindrical (BHS14 Corollary 14.4)
    Used in Proposition 4.5 to get free subgroups.
  • standard math Theorem 2.28 (consistent tuples and the quasi-inverse of projection)
    Used in Proposition 2.27 product decomposition.

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Pith. "Pith review of Hierarchically hyperbolic groups and uniform exponential growth." pith.science (2026). https://pith.science/paper/T4A6DTH4

@misc{pith2026190900439,
  author       = {Pith},
  title        = {Pith review of: Hierarchically hyperbolic groups and uniform exponential growth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4A6DTH4}},
  note         = {Machine review of arXiv:1909.00439}
}
read the original abstract

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterization of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature. In particular, we obtain the first proof of uniform exponential growth for certain groups that act geometrically on CAT(0) cubical spaces of dimension 3 or more. Under additional hypotheses, we show that a quantitative Tits alternative holds for hierarchically hyperbolic groups.

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