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Some remarks on decay in countable groups and amalgamated free products

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Pith's one-line read Amalgamated free products preserve rapid and subexponential decay whenever the factor length functions agree on the amalgam or their distortion is controlled, with explicit decay bounds and sharp optimality examples.

desk verdict Introduces subexponential decay and proves solid permanence results for amalgamated free products; core math holds up, but the introduction overclaims on Grigorchuk's group and Section 3 hides a small but real assumption about length functions. read the letter →

arxiv 2509.08754 v1 pith:WJZLLSLD submitted 2025-09-10 math.GR math.FAmath.OA

Srivatsav Kunnawalkam Elayavalli , Gregory Patchell , Lizzy Teryoshin This is my paper · ORCID
classification math.GRmath.FAmath.OA MSC 46L0520E0620F65
keywords subexponentialdecayrapidamalgamatedfreeproductslengthfunctionsuniversaldistortiongraphC*-algebras
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 argues that subexponential decay (SD), the weakening of rapid decay in which the operator norm of a finitely supported function on a group is bounded by a subexponential function of its support radius rather than a polynomial one, is a natural and useful property for countable groups. It proves SD is preserved under subgroups, direct products, free products, and graph products, and it constructs a group with SD but not RD whose amenable subgroups all have polynomial growth. The central theorem is a permanence result for amalgamated free products: if two groups with RD or SD are glued along a common subgroup on which their length functions agree, the amalgam inherits the decay property, and the same holds without exact agreement when the universal length function is only mildly distorted. The paper also shows these distortion hypotheses are essentially optimal, since explicit amalgams such as $SL_2(\mathbb{Z}[1/p])$ and $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ have exponential length distortion and fail RD and SD.

What carries the argument

The load-bearing object is the length function on the amalgamated free product and its interaction with reduced words. When $L_G$ and $L_H$ agree on $A$, the paper defines $L(k)=\min\{L_A(a)+L_G(g_1)+L_H(h_1)+\cdots\}$ over reduced decompositions, proves it is subadditive and symmetric (Lemma 3.3), and shows it restricts to the original lengths on $G$ and $H$. When they do not agree, the universal length $L_U(k)=\min\{\sum_i L_i(k_i): \prod_i k_i=k,\ k_i\in G\cup H\}$, in which an element of $A$ may be measured by either factor's length, creates agreeing restrictions. The decay proof slices group-ring elements by syllable count and length, chooses $L$-minimizing coset representatives for $A$, and applies the RD or SD inequality inside $A$, $G$, and $H$ in the four cancellation regimes inherited from the original free-product proof; Proposition 2.10 then packages the per-shell estimates into a global decay function.

What would settle it

Exhibit an amalgamated free product $G*_A H$ in which $G$ and $H$ have RD, the pairs $(L_U|_G,L_G)$ and $(L_U|_H,L_H)$ are subexponentially distorted, but $G*_A H$ contains a finitely generated amenable subgroup with exponential growth; by the paper's own Proposition 2.8 such a group cannot have SD, which would refute Theorem B(2).

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Extended reading notes

Core claim

The paper's central claim is that subexponential decay is preserved by amalgamated free products under controlled length distortion. Theorem A states that if $(G,L_G)$ and $(H,L_H)$ have RD (respectively SD) and $L_G|_A = L_H|_A$ on a common subgroup $A$, then $\Gamma = G*_A H$ has RD (respectively SD) with respect to $L(k) = \min\{L_A(a)+L_G(g_1)+L_H(h_1)+\cdots\}$, and if both factors have $f$-decay then the amalgam has $(2x+1)^{7/2}f(2x)$-decay. Theorem B replaces exact agreement by the universal length $L_U$, defined by minimizing sums of factor lengths over decompositions in $G\cup H$, and says the amalgam has RD when both factors do and the pairs $(L_U|_G,L_G)$ and $(L_U|_H,L_H)$ are polynomially distorted, and has SD when the distortion is subexponential with RD factors or linear with SD factors. Proposition C and Corollary D transfer these hypotheses to the distortion of the two lengths on the amalgam $A$ itself, with logarithmic distortion preserving RD and sublinear distortion preserving SD. The paper completes the picture by showing that the amalgamated decompositions of $SL_2(\mathbb{Z}[1/p])$ and $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ have exponential distortion, so the distortion assumptions in the permanence theorem cannot simply be dropped.

Load-bearing premise

The Section 3 proofs assume the length functions are integer-valued and have finite balls, so that coset representatives can be chosen to minimize $L$ and the number of syllables is bounded by $L_U(g)$; if a length function is merely real-valued or not proper, those minimizations and finite-ball arguments can break down, and the paper justifies the assumption only by saying such lengths can be assumed as in [20].

Editorial extensions

If this is right

  • Any amalgamated free product of two RD groups over a common subgroup on which the two length functions agree is RD; this covers arbitrary group doubles of RD groups and yields a new inductive proof that graph products of RD groups are RD.
  • Without exact agreement, RD still passes if the universal length is polynomially distorted on both factors, and SD passes if the distortion is subexponential with RD factors or linear with SD factors.
  • If the two length functions differ on the amalgam only by a logarithmic term, RD passes to the product; a sublinear difference passes SD.
  • When both factors admit $f$-decay and the lengths agree on the amalgam, the amalgam admits $(2x+1)^{7/2}f(2x)$-decay, so the decay function degrades only by a fixed polynomial factor.
  • The reduced free product $C^*$-algebra $C^*_r(G*H)$ is selfless whenever $G$ and $H$ have SD, $G$ has a torsion-free element, and $H$ is infinite, recovering selflessness for free products involving intermediate-growth groups.

Reading between the lines

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

  • The same Khintchine-based mechanism that proves selflessness for free products may extend to amalgamated free products over subgroups, since Theorem B's distortion control could supply the needed length bounds; testing selflessness of $C^*_r(G*_A H)$ under linear distortion is a natural next step.
  • The two exponential-distortion examples suggest that preservation of decay is governed by the distortion of the universal length rather than by the size or amenability of the amalgam; if so, the boundary between RD and SD in amalgams can be read off from a single length-function comparison.
  • The SD-without-RD example is built with a specially engineered length function on an infinitely generated group; a finitely generated word-length example with the same features would show the phenomenon is not an artifact of the construction, and would sharpen the open question the paper records.
  • The Fréchet subalgebra construction of Section 2.6 may still be useful for concrete SD groups even though holomorphic closure fails for general subexponential $f$; checking inverse-closedness on the constructed SD example would test whether the obstruction is real or only a limitation 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.

Referee Report

2 major / 5 minor

Summary. This paper studies subexponential decay (SD), a weakening of rapid decay (RD) for countable groups equipped with length functions. It proves several permanence results: SD passes to subgroups, direct products, free products, and graph products; it constructs a countable group with SD but not RD whose amenable subgroups have polynomial growth; it proves a selflessness result for reduced free product C*-algebras; and it proves that RD/SD pass to amalgamated free products when the factor length functions agree on the amalgam, with explicit decay functions (Theorem 3.6), plus a general distortion version via a universal length function (Theorem 3.8, Proposition 3.9, Corollary 3.10). The paper also gives explicit examples showing that the distortion hypotheses are necessary.

Significance. If the proofs are completed, the main value is the elementary, local proof of amalgamated free product permanence for both RD and SD with explicit constants, complementing the independent work of Chatterji-Gautero, and the formulation of SD as a useful weakening of RD. The optimality examples involving SL2(Z[1/p]) and Z^2⋊SL2(Z) are concrete and well chosen, and the paper is transparent about overlap with [4] and about the limitations in Remark 2.19. However, the manuscript currently has a genuine gap in the reduction to integer-valued length functions and an incorrect advertised application to Grigorchuk's group; these need to be fixed before the results can be accepted in the present form.

major comments (2)
  1. [Section 3 and Definition 2.3] The unproved assertion at the start of Section 3, 'We may assume that L_G, L_H are integer-valued as in [20]', is load-bearing and does not follow from Definition 2.3, which allows arbitrary R_{\ge 0}-valued length functions. Lemma 3.5 chooses minimizers of L(g_1 a) over a\in A (Case I) and of L(aw) over w\in T (Case II); these minima need not be attained for real-valued lengths. The inequality K(g)\le L(g) used throughout Lemmas 3.4 and 3.5, and the bound n\le L_U(g) in Proposition 3.9, require every nonidentity syllable to have length at least 1. The same issue affects Section 2: Proposition 2.10 decomposes \phi as \sum \phi_k over spheres C_k=\{g:l(g)=k\} with k\in\mathbb{N}, so it silently assumes integer-valued lengths. The authors should state and prove the reduction once: for any length function l, l'(e)=0 and l'(g)=\max(1,\lceil l(g)\rceil) for g\ne e is a length function, agrees with l on any common subgroup where l_G|_A=l_H|_A, only shrinks balls, and hence preserves f-decay. With this replacement all minimizations and inequalities cited above are justified; without it the proof of Theorem 3.6 is incomplete as written.
  2. [Introduction, p. 2, and Theorem 2.16] The introduction states that Theorem 2.16 'recovers selflessness for C^*_r(G*G) where G is the well-known Grigorchuk group.' But Grigorchuk's group is a 2-group, so every element has finite order; hence the hypothesis of Theorem 2.16 that 'G has a torsion free element' is not satisfied by either factor. As stated, the theorem does not apply to G*G for Grigorchuk's group. The application can be repaired either by weakening the hypothesis to the natural condition that the free product contain an infinite-order element (which holds for any nontrivial free product of nontrivial groups) or by citing a version of [1, Proposition 3.1] that does not require a torsion-free factor; otherwise the advertised corollary should be removed.
minor comments (5)
  1. [Section 1, first paragraph] The first paragraph defines a length function as a map l:G\to\mathbb{N}, while Definition 2.3 allows R_{\ge 0}-valued lengths; this inconsistency should be resolved by adopting the integer-valued convention after the reduction in my first major comment is stated and proved.
  2. [Section 2.4, Proposition 2.15] The proof of Proposition 2.15 delegates several key steps to 'following arguments in Section 6 of [8]' without stating the polynomial Q or the precise clique sums; please include the complete argument or clearly label this part as a reduction to [8], since as written it is difficult to verify independently.
  3. [Section 3, Lemma 3.5, Case IV] In the displayed inequality after 'We therefore have', the expression '\varphi'(g_1)*\psi'(g_1) g_2 (t)' contains a spurious subscript '(g_1)'; it should read \psi'_{g_2}(t).
  4. [Section 3, Lemma 3.5, Case IV] In the final paragraph of Case IV, 'K(w^{-1}vg_2)=K(w'^{-1}v'g'_2)=kthen' should read '=k then'.
  5. [Section 2.5, proof of Theorem 2.16] The notation |\varphi_n(t)|=4n+1 is used without definition; please specify the word length with respect to which this value is measured.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main permanence theorems are proved directly; self-citations are auxiliary, not inputs to the central derivation.

full rationale

The central derivation chain is self-contained rather than circular. Theorem A (Theorem 3.6) is proved by constructing the amalgamated length L in Definition 3.1, showing in Lemmas 3.2-3.3 that it is a length function extending L_G and L_H, and then proving the convolution estimate in Lemma 3.5 directly from the RD/SD hypotheses on G, H, and A. No parameter is fitted and the estimate f(2k') is obtained from the given decay functions of the factors. Theorem B (Theorem 3.8) is a genuine reduction to Theorem A via the universal length L_U and distortion bounds, and Proposition C proves the distortion estimates from the hypothesis that L_G|A and L_H|A are f-distorted. These are valid reductions, not definitions that smuggle in the desired conclusion. The self-citations, notably [1], [10], and [18], are used only in the auxiliary selflessness application in Section 2.5, where Proposition 3.1 and 3.2 of [1] and Lemma 4.4 of [18] are cited as external supporting results, not as the target permanence theorem; the proof adds a new argument via Lemma 2.17. The acknowledged overlap with Chatterji-Gautero [4] is explicitly disclosed and does not make the independent proof circular. The one substantive proof gap is at the opening of Section 3, where the paper says 'We may assume that L_G, L_H are integer-valued as in [20]' after Definition 2.3 allowed arbitrary nonnegative real-valued lengths. This is a missing justification, not a circularity: it does not identify an input with an output or rename a fitted parameter as a prediction. A straightforward repair (replace l by max(1, ceil(l)) for nontrivial elements) shows the gap is fixable and does not affect the independence of the derivation. Therefore no circular step is identified; the score of 2 reflects only the presence of self-citations in parts of the paper, none of which is load-bearing for the main theorems.

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

No free parameters and no invented entities: the paper introduces definitions (SD, L_U) rather than entities requiring independent falsifiable evidence. The axioms are mostly standard borrowed results; the most fragile is the implicit properness of length functions.

assumptions (6)
  • standard math Ricard-Xu Khintchine-type inequality (Lemma 4.4 of [18], [29]) provides the norm bound in Lemma 2.17.
    Used without proof as a black box in the proof of Theorem 2.16.
  • standard math Proposition 3.1 of [1] yields homomorphisms phi_n: G*H*<t> -> G*H that are injective on B_n and stretch t to length 4n+1.
    Borrowed from prior work by Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell; the present paper does not prove it and does not state its hypotheses.
  • standard math Jolissaint's Proposition 1.2.6 [20] characterizes RD/SD through layerwise convolution estimates.
    Used in Proposition 2.10 and Lemma 3.4 as the engine that converts layerwise bounds into a global decay function.
  • domain assumption Length functions under consideration are proper and integer-valued (l(g) = 0 iff g = e).
    Assumed implicitly ('We may assume that L_G and L_H are integer-valued as in [20]') and used for finite balls, minimizations in Lemma 3.5, and n <= L_U(g) in Proposition 3.9.
  • domain assumption The Grigorchuk group has subexponential intermediate growth and is a torsion group.
    Used in Remark 2.7 and in the introduction's claimed application to C*_r(G*G); the torsion property contradicts the stated hypothesis of Theorem 2.16.
  • domain assumption SL2(Z[1/p]) decomposes as an amalgamated free product of two copies of SL2(Z) over a finite-index subgroup A and contains exponential-growth amenable subgroups.
    Used in the optimality discussion to show exponential distortion kills RD/SD.

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Pith. "Pith review of Some remarks on decay in countable groups and amalgamated free products." pith.science (2026). https://pith.science/paper/WJZLLSLD

@misc{pith2026250908754,
  author       = {Pith},
  title        = {Pith review of: Some remarks on decay in countable groups and amalgamated free products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJZLLSLD}},
  note         = {Machine review of arXiv:2509.08754}
}
abstract

In this note, we first study the notion of subexponential decay (SD) for countable groups with respect to a length function, which generalizes the well-known rapid decay (RD) property, first discovered by Haagerup in 1979. Several natural properties and examples are studied, especially including groups that have SD, but not RD. This consideration naturally has applications in $C^*$-algebras. We also consider in this setting a permanence theorem for decay in amalgamated free products (proved also recently by Chatterji--Gautero), and demonstrate that it is in a precise sense optimal.

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