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Dynamic asymptotic dimension growth for group actions and groupoids

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a bounded-geometry metric space and its coarse groupoid share the same dimension growth, and that subexponential ($x^\alpha$, $\alpha<1$) growth forces amenability.

desk verdict Interesting new invariant and several good results, but the proof of the main equivalence in Theorem 4.16 has a real gap in one direction. read the letter →

arxiv 2411.19712 v2 pith:ZAANVZPX submitted 2024-11-29 math.DS math.FAmath.GR

Hang Wang , Yanru Wang , Jianguo Zhang , Dapeng Zhou This is my paper · ORCID
classification math.DSmath.FAmath.GR MSC 54F4522A2237A5537B0546L05
keywords dynamicasymptoticdimensiongrowthgroupactionsétalegroupoidscoarsegroupoidamenabilitypartitionofunitypropertyA
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 introduces a quantitative version of dynamic asymptotic dimension, tracking how the minimal number of cover pieces needed to control arrows of bounded length grows as the scale increases. The central result is an equivalence: for any discrete metric space $X$ of bounded geometry, the asymptotic dimension growth of $X$ is the same growth type as the dynamic asymptotic dimension growth of its coarse groupoid $G(X)$. From this, the paper shows that subexponential dynamic asymptotic dimension growth of $G(X)$ implies that $G(X)$ is amenable. A more general theorem states that every $\sigma$-compact étale groupoid with compact unit space and dynamic asymptotic dimension growth at most $x^\alpha$ ($0<\alpha<1$) is amenable, providing a dimension-based route to amenability for groupoids and many concrete examples.

What carries the argument

The central objects are the dynamic asymptotic dimension function $\mathrm{dad}_G$ of an étale groupoid (the minimal number of open sets in a cover of the unit space such that all arrows of length $<R$ with source and range in one set lie in a precompact subgroupoid) and the coarse groupoid $G(X)$ of a bounded-geometry metric space, whose arrows encode proximity at all scales. The argument's load-bearing identity is $\mathrm{ad}_X \asymp \mathrm{dad}_{G(X)}$ (Theorem 4.16), obtained by translating the coarse-geometric formulation of asymptotic dimension growth into controlled separations inside $G(X)$. For amenability, the key mechanism is the generalized partition of unity of Proposition 5.1: a growth bound $f\le x^\alpha$ yields continuous functions $(\varphi_i)$ on the unit space with $\sum_i \varphi_i^p = 1$ and uniformly small variation along arrows of length $<R$, from which an almost invariant measure is assembled; here $p$ is an integer depending only on $\alpha$.

What would settle it

Check the displayed inequality (5.1) in Proposition 5.1: as written the left-hand side lacks division by the auxiliary scale $N(R)$. With the natural correction, test whether the constant $c$ asserted to exist in Remark 5.7 actually makes the corrected inequality hold for the model function $f(x)=\lceil x^{1/2}\rceil$ at $R=1$ and $\varepsilon=0.1$; if no such $c$ exists, or if some admissible $f\le x^\alpha$ violates the corrected estimate, the proof of Theorem 6.4 does not go through.

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Extended reading notes

Core claim

The paper's central claim is that the growth type of the dynamic asymptotic dimension function—the minimal $m$ such that for each scale $R$ the arrows of length $<R$ with source and range in a single cover element lie in a precompact subgroupoid—is a meaningful quantitative invariant, not just a finiteness test. The load-bearing equivalence is Theorem 4.16: for a discrete metric space $X$ of bounded geometry with coarse groupoid $G(X)$ and any nondecreasing $f$, $\mathrm{ad}_X \asymp f$ if and only if $\mathrm{dad}_{G(X)} \asymp f$. A direct corollary is that subexponential DAD growth of $G(X)$ implies amenability, matching the known implication from subexponential asymptotic dimension growth to property A. The paper then proves that any $\sigma$-compact étale groupoid with compact unit space and DAD growth at most $x^\alpha$ for some $0<\alpha<1$ is amenable; the proof constructs, from the growth bound, a generalized partition of unity whose $p$-th powers sum to one and whose variation along short arrows is uniformly small, and this yields an almost invariant measure.

Load-bearing premise

The proof assumes that a function that grows no faster than x to the alpha power (with alpha less than 1) still grows slowly enough that an error term involving the function divided by a large auxiliary scale can be made arbitrarily small; the paper only checks this by a sketch for one representative function, not for every admissible function.

Editorial extensions

If this is right

  • For every bounded-geometry metric space, asymptotic dimension growth and the dynamic asymptotic dimension growth of the associated coarse groupoid have the same growth type, so results can flow across the coarse-groupoid dictionary.
  • Subexponential dynamic asymptotic dimension growth of a coarse groupoid forces the groupoid to be amenable, yielding new amenable groupoids from spaces such as geodesic coarse median spaces of finite rank and at most exponential volume growth.
  • Every $\sigma$-compact étale groupoid with compact unit space and dynamic asymptotic dimension growth at most $x^\alpha$ ($0<\alpha<1$) is amenable, a strictly quantitative generalization of the finite-dimension amenability theorem that removes the freeness hypothesis.
  • A countable discrete group with subexponential asymptotic dimension growth admits a free, minimal action on the Cantor set whose dynamic asymptotic dimension growth is at most the same function, so examples like the wreath product $\mathbb{Z}\wr\mathbb{Z}$ give actions with polynomial dynamic asymptotic dimension growth.

Reading between the lines

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

  • If the apparent missing factor in inequality (5.1) is a typographical slip, the amenability theorem likely extends beyond $x^\alpha$ to any subexponential growth function by refining the partition-of-unity estimates; if it is not, there may exist non-amenable groupoids with subexponential DAD growth, which would be a surprising separation.
  • The equivalence theorem suggests an untapped two-way bridge: dimension-growth bounds proved for group actions could be transplanted to coarse spaces, and coarse-geometric constructions could produce actions with prescribed growth.
  • The partition-of-unity proof has the shape of a quantitative Reiter condition, so the theorem may yield explicit estimates on the almost invariant measures in terms of the growth function, not just qualitative amenability.
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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 / 5 minor

Summary. The paper introduces the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces and for locally compact étale groupoids, generalizing the finite dynamic asymptotic dimension of Guentner–Willett–Yu. It establishes several equivalent definitions of asymptotic dimension growth, connects asymptotic dimension growth of a bounded-geometry discrete metric space to the dynamic asymptotic dimension growth of its coarse groupoid (Theorem 4.16), derives amenability results (Corollary 4.19 and Theorem 6.4), and uses Bartels–Lück–Reich actions to construct group actions with prescribed growth behavior. The central claimed contributions are the equivalence between coarse and dynamic growth (Theorem 4.16) and the amenability of groupoids with slow dynamic asymptotic dimension growth (Theorem 6.4).

Significance. If the main results hold, the paper provides a quantitative dimension invariant that unifies coarse geometry, dynamical systems, and operator algebras, extending known finite-dimensional theorems to a growth setting. The paper is well structured, and the development of three equivalent definitions of asymptotic dimension growth (Section 2) and the partition-of-unity machinery (Section 5) are potentially useful tools. The authors are also careful to cite and build on prior work (GWY17, ANWZ18, STY02, MW20), and the paper does not introduce fitted parameters or circular reasoning. However, the proof of the central equivalence Theorem 4.16 contains a substantial gap, and the auxiliary inequality in Proposition 5.1 is misstated; both issues must be resolved before the results can be accepted.

major comments (2)
  1. [Section 4.2, proof of Theorem 4.16] The second half of the proof aims to show dad_{G(X)} ≼ |ad_X. It defines U_i := ⊔_{U∈U_i} \overline{U} in βX and asserts that {U_0, ..., U_{|ad_X(R)|}} is a cover of βX because U covers X. This is false. Since X has bounded geometry, each bounded U is finite, so \overline{U} = U. The union over a family U_i is then just the subset of X covered by that family; it need not be open in βX, and the union over all i is exactly X, not βX. For example, with X=Z and the cover by singletons decomposed into evens and odds, the two sets are the even and odd integers, whose union is Z, a proper dense open subset of βZ. Consequently, the construction of the precompact subgroupoids G_i does not yield the required open cover of the unit space of G(X), and the inequality dad_{G(X)} ≼ |ad_X is not established. This gap directly affects Corollary 4.19 and the group-action consequences in Section 4.3.
  2. [Section 5, equation (5.1)] The displayed inequality (5.1) omits a division by N(R) in the first term. As written, the left-hand side contains the term 2p(f(N(R)R+1)+1), which, under f ≼ x^α, grows like N(R)^α and cannot be made smaller than ε by increasing the constant c. The estimates in equations (5.6)–(5.8) actually give the bound (2p(f(N(R)R+1)+1) + 2p(f(N(R)R+1)+1)^{1/p+1})/N(R), so the correct version of (5.1) should have the first term divided by N(R). With this correction, the existence of c is plausible because f(N(R)R+1) = O(N(R)^α) and α<1, but the authors do not prove this for arbitrary f in the given growth class; Remark 5.7 only treats the concrete function f(x)=⌈x^α⌉. Since Proposition 5.1 is the key tool for Theorem 6.4, a complete argument for general f ≼ x^α is required.
minor comments (5)
  1. [Throughout] The notation for the coarse asymptotic dimension function is inconsistent: Lemma 2.13 defines it as }ad_X, while the proof of Theorem 4.16 uses |ad_X. Please use one symbol consistently.
  2. [Section 2.1, proof of Proposition 2.5] There is a typo: 'unifromly' should be 'uniformly'. Also, the expression 'ĆadX' appears to be a misprint for the function defined earlier.
  3. [Section 5, Proposition 5.1 statement] The indexing of the cover is written as f(r(R+1)csR+1), which is ambiguous; it should clearly indicate the floor/ceiling notation, e.g., f(⌈(R+1)c⌉ R+1). The same notational issue appears in the proof of Proposition 5.1 and in Theorem 6.4.
  4. [Section 3.2, Proposition 3.13] The word 'equivaraint' should be 'equivariant'.
  5. [Section 4.3, Theorem 4.22] The word 'metrizible' should be 'metrizable'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are proved from definitions and external theorems; the proof gaps noted by the reviewer are correctness issues, not circularity.

full rationale

I walked the derivation chain. Asymptotic dimension growth (Definitions 2.1, 2.13) and dynamic asymptotic dimension growth for actions and groupoids (Definitions 3.2, 4.6) are introduced as independent quantitative invariants, and the main equivalence Theorem 4.16 is proved from these definitions using the coarse groupoid construction of Skandalis-Tu-Yu; neither half of the proof feeds the target statement back in as an input. Corollary 4.19 and Theorem 6.4 rely on external results (Ozawa's subexponential-growth-implies-property-A theorem, the STY amenability equivalence for coarse groupoids, Ma-Wu coarse length functions, and Guentner-Willett-Yu partition-of-unity techniques), not on the paper's own claims. The only self-citations are to the authors' earlier Lp-Roe-algebra papers and are purely background, not load-bearing. I found no step in which a 'prediction' or 'first-principles result' is equivalent to its inputs by construction, and no parameter fitted to a subset of data is later presented as a prediction. The second half of Theorem 4.16 does contain a genuine mathematical gap: closures of bounded subsets of X need not cover the Stone-Cech unit space beta X, so the displayed cover construction does not follow as written. That is a correctness defect, not circularity. Likewise, the existence of the constant c in Proposition 5.1 is only sketched for a special function f, which is an incomplete verification rather than a circular definition. Under the stated rules, the honest finding is no significant circularity, score 0.

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

No free parameters or invented entities are introduced. The paper relies on standard mathematical background and several cited theorems as assumptions, which are all explicitly stated in the text.

assumptions (4)
  • domain assumption Existence and uniqueness up to coarse equivalence of coarse continuous length functions on sigma-compact locally compact Hausdorff etale groupoids (Theorem 4.4, from Ma-Wu [MW20]).
    Used in Definition 4.6 and in the proof of Theorem 6.4 to choose the length function ell.
  • domain assumption Subexponential asymptotic dimension growth implies Yu's property A (Ozawa [Oza12], Oppenheim [Opp14]; quoted as Theorem 4.18).
    Bridges asymptotic dimension growth to amenability of the coarse groupoid in Corollary 4.19 and Theorem 4.21.
  • domain assumption Property A for a discrete metric space is equivalent to amenability of its coarse groupoid (Skandalis-Tu-Yu, cited before Corollary 4.19).
    Essential for Corollary 4.19 and the interpretation of the main equivalence.
  • domain assumption Finite dynamic asymptotic dimension groupoids are amenable via Proposition 5.8 (adapted from GWY17) and open precompact subgroupoids are amenable (Lemma 6.2, from GWY24).
    Used in the proof of Theorem 6.3.

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Pith. "Pith review of Dynamic asymptotic dimension growth for group actions and groupoids." pith.science (2026). https://pith.science/paper/ZAANVZPX

@misc{pith2026241119712,
  author       = {Pith},
  title        = {Pith review of: Dynamic asymptotic dimension growth for group actions and groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZAANVZPX}},
  note         = {Machine review of arXiv:2411.19712}
}
abstract

We introduce the notion of dynamic asymptotic dimension growth for actions of discrete groups on compact spaces, and more generally for locally compact \'etale groupoids. Using the work of Bartels, L\"uck, and Reich, we bridge asymptotic dimension growth for countable discrete groups with our notion for their group actions, thereby providing numerous concrete examples. Moreover, we demonstrate that the asymptotic dimension growth for a discrete metric space of bounded geometry is equivalent to the dynamic asymptotic dimension growth for its associated coarse groupoid. Consequently, we deduce that the coarse groupoid with subexponential dynamic asymptotic dimension growth is amenable. More generally, we show that every $\sigma$-compact locally compact Hausdorff \'etale groupoid with compact unit space having dynamic asymptotic dimension growth at most $x^{\alpha}$ $(0<\alpha<1)$ is amenable.

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