{"id":"c6620dcd-6534-43ad-b224-215e08f204ae","arxiv_id":"2509.08754","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Subexponential decay (SD) is introduced as a weakening of rapid decay; permanence of SD/RD under amalgamated free products is proved with sharp distortion bounds, with applications to selflessness of reduced C*-algebras.","lead":"This note studies subexponential decay (SD), a weaker version of the classical rapid decay property for countable groups, and proves that SD and RD are preserved under amalgamated free products under controlled length-function distortion. It also applies SD to reduced C*-algebras, including a selflessness result for free products and an explicit example of an SD group without RD.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3 silently assumes length functions are integer-valued and positive; the main proof's minimizations and K≤L step require this, and the reduction from Definition 2.3 is not justified.","rationale":"The reader correctly targets the Section 3 length-function assumption. The concern lands as a missing justification, not as a counterexample: the replacement l'(g)=max(1,⌈l(g)⌉) preserves decay and agreement on A, so the proof can be repaired in a few lines. Properness (finite balls) is not actually used: after the replacement, minima exist because values are non-negative integers, and the sums over representatives are finite because test functions have finite support. The separate Grigorchuk claim in the introduction is an error in an application, but it does not affect the correctness of Theorems A/B. Hence the reader's conditional verdict should stand.","tokens_in":25538,"tokens_out":47954,"duration_ms":785437,"concrete_test":"Verify the proposed replacement in the setting of Lemma 3.5: define l'(g)=max(1,⌈l(g)⌉) for g≠e, l'(e)=0; check (i) subadditivity, (ii) B'_{n}⊆B_n so any f-decay for (G,l) is f-decay for (G,l'), and (iii) L_G|_A=L_H|_A implies L'_G|_A=L'_H|_A. Then re-run the four cases of Lemma 3.5 and Proposition 3.9 with l' in place of l, confirming that the minima over A are attained and K(g)≤L(g), n≤L_U(g) hold. If all steps pass, the flagged assumption is a harmless omitted justification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is the unproved passage at the start of Section 3: 'We may assume that L_G, L_H are integer-valued as in [20].' Definition 2.3 permits arbitrary R_{≥0}-valued lengths. Lemma 3.5 chooses A-coset representatives minimizing L(g_1 a) over a∈A; if the length is real-valued the minimum need not be attained. Likewise K(g)≤L(g) and the later bound n≤L_U(g) in Proposition 3.9 require that non-identity syllables have positive integer length. The proof does not establish that lengths can be made integer-valued and positive without changing the hypotheses. Properness is not the real issue: the repair is to replace l by l'(g)=max(1,⌈l(g)⌉) for g≠e and l'(e)=0. This is still a length function, balls only shrink, so f-decay is preserved, and equality on A is preserved. With this replacement all minimizations in Lemma 3.5 and the inequality K≤L are justified. The paper's central theorems therefore appear sound, but the text needs this one-line justification; as written the reduction is a genuine gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25770,"tokens_out":14027,"duration_ms":125581,"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":[{"comment":"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.","section":"Section 3 and Definition 2.3"},{"comment":"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.","section":"Introduction, p. 2, and Theorem 2.16"}],"minor_comments":[{"comment":"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.","section":"Section 1, first paragraph"},{"comment":"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.","section":"Section 2.4, Proposition 2.15"},{"comment":"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).","section":"Section 3, Lemma 3.5, Case IV"},{"comment":"In the final paragraph of Case IV, 'K(w^{-1}vg_2)=K(w'^{-1}v'g'_2)=kthen' should read '=k then'.","section":"Section 3, Lemma 3.5, Case IV"},{"comment":"The notation |\\varphi_n(t)|=4n+1 is used without definition; please specify the word length with respect to which this value is measured.","section":"Section 2.5, proof of Theorem 2.16"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a useful paper. It defines subexponential decay (SD), shows it is strictly weaker than RD via a Sapir-type construction, and proves that SD/RD survive amalgamated free products under controlled distortion. The main theorems look right. Two things need fixing before I'd want to cite it: the introduction claims a Grigorchuk-group application that doesn't satisfy the theorem's hypotheses, and Section 3 assumes without comment that length functions can be taken integer-valued.\n\nWhat's genuinely new: SD as a named property, the existence of SD-without-RD groups with no superpolynomial amenable subgroups (Theorem 2.13), and the distortion-based permanence results (Theorem B parts 2 and 3, Proposition C, Corollary D) with the optimality examples (SL2(Z[1/p]) and Z^2 ⋊ SL2(Z)). Theorem A is independently proved and new for SD, though the RD case largely overlaps with Chatterji–Gautero; the authors say so plainly. The proof style is a plus: it's local and combinatorial, not black-box Jolissaint. The selflessness application is real but now mostly subsumed by Ozawa's much broader theorem; again, they acknowledge that.\n\nThe problems. First, the introduction says Theorem 2.16 'in particular recovers selflessness for C*_r(G*G) where G is the well known Grigorchuk's group.' But Grigorchuk's group is torsion, and Theorem 2.16 requires one factor to contain an infinite order element. That sentence is simply wrong and should be removed or replaced with a valid example. Second, Section 3 opens with 'We may assume that L_G, L_H are integer-valued as in [20].' The stress-test note is right: Definition 2.3 allows real-valued lengths, and the minimizations in Lemma 3.5 plus the bound n ≤ L_U(g) in Proposition 3.9 require positive integer values. The repair is easy—replace l(g) with max(1, ⌈l(g)⌉) for g≠e, l(e)=0—but as written it's a genuine gap. Since the fix is one line, this shouldn't hold up publication, but the authors need to say it.\n\nThe gaps are in presentation, not in the central arguments. The four cases in Lemma 3.5 are carefully done, the distortion examples check out, and the decay estimates are explicit. The paper is honest about its limitations, including the open question about SD replacing RD in the Baum–Connes context. I'd bring it to a reading group interested in decay properties or reduced C*-algebras. It's not a transformative paper, but it's a solid, reusable contribution.\n\nRecommendation: send it to peer review. With the two corrections above, I'd accept it.","headline":"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.","tokens_in":26327,"tokens_out":10463,"would_cite":true,"duration_ms":87622,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05","20E06","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["subexponential decay","rapid decay","amalgamated free products","length functions","universal length","distortion","graph products","C*-algebras"],"falsifier":"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).","tokens_in":25355,"feed_emoji":"🔗","tokens_out":14102,"duration_ms":113644,"temperature":0.7,"pith_summary":"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.","feed_headline":"Amalgamated free products keep rapid and subexponential decay","feed_subtitle":"Gluing two groups along a common subgroup preserves their decay property when length functions are not too distorted.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the RD/SD framework and the convolution criterion (Proposition 1.2.6) that converts per-shell estimates into decay; also the source of the integer-valued length assumption and the finite/central-index amalgam cases.","marker":"[20]"},{"why":"Provides the original free-product RD argument whose four-case cancellation analysis the amalgam proof adapts in structure.","marker":"[16]"},{"why":"Independently proves a graph-of-groups RD permanence whose Proposition 1.3 overlaps with Theorem B(1); the paper uses it to position its independent proof and its SD extensions.","marker":"[4]"},{"why":"Gives the construction that the paper modifies to produce an SD group without RD whose amenable subgroups all have polynomial growth.","marker":"[31]"},{"why":"Establishes RD permanence for graph products; the paper adapts the argument with adjusted constants to obtain the SD version.","marker":"[8]"},{"why":"Supplies the noncommutative Khintchine-type inequalities used in Lemma 2.17 to replace RD by SD in the selflessness proof.","marker":"[29]"},{"why":"Provides the lemma bounding alternating-word convolutions in free products by the factors' SD decay, the key step in Theorem 2.16.","marker":"[18]"},{"why":"Motivates the explicit exponential-distortion example in $\\mathbb{Z}^2 \\rtimes SL_2(\\mathbb{Z})$ through its discussion of cyclic subgroups with exponential growth and word-length distortion.","marker":"[26]"}],"fun_headline_variants":["Gluing groups preserves decay if lengths are barely distorted","Amalgamated free products retain decay under controlled lengths","Subexponential decay survives amalgamation with mild distortion","Decay permanence in amalgams: distortion bounds are sharp"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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].","fun_headline_variants_meta":{"raw":{"variants":["Gluing groups preserves decay if lengths are barely distorted","Amalgamated free products retain decay under controlled lengths","Subexponential decay survives amalgamation with mild distortion","Decay permanence in amalgams: distortion bounds are sharp"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3228,"prompt_tokens":970,"completion_tokens":2258,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":2191}},"tokens_in":586,"tokens_out":2258,"duration_ms":17219,"temperature":1.0,"reasoning_tokens":2191,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:00:49.409066+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the RD/SD framework and the convolution criterion (Proposition 1.2.6) that converts per-shell estimates into decay; also the source of the integer-valued length assumption and the finite/central-index amalgam cases."},{"cited_title":"Math.50(1978), no","cited_arxiv_id":null,"evidence_quote":"Provides the original free-product RD argument whose four-case cancellation analysis the amalgam proof adapts in structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Independently proves a graph-of-groups RD permanence whose Proposition 1.3 overlaps with Theorem B(1); the paper uses it to position its independent proof and its SD extensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the construction that the paper modifies to produce an SD group without RD whose amenable subgroups all have polynomial growth."},{"cited_title":"Holt, and Sarah Rees,Rapid decay is preserved by graph products, J","cited_arxiv_id":null,"evidence_quote":"Establishes RD permanence for graph products; the paper adapts the argument with adjusted constants to obtain the SD version."},{"cited_title":"Reine Angew","cited_arxiv_id":null,"evidence_quote":"Supplies the noncommutative Khintchine-type inequalities used in Lemma 2.17 to replace RD by SD in the selflessness proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates the explicit exponential-distortion example in $\\mathbb{Z}^2 \\rtimes SL_2(\\mathbb{Z})$ through its discussion of cyclic subgroups with exponential growth and word-length distortion."}],"review_version":2}