{"id":"acb479a0-f217-4f5a-bfec-1047e961e5d6","arxiv_id":"2412.04932","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Trickle groups unify right-angled Artin/Coxeter groups, cactus groups, Thompson group F, and ordered quandle groups, and they all inherit a terminating and confluent rewriting system and a solution to the word problem.","lead":"This paper introduces trickle groups, a family of groups defined by relations of the form xy=zx and x^mu=1, and proves they have terminating and confluent rewriting systems, normal forms, and a solvable word problem. The family includes right-angled Artin/Coxeter groups, cactus groups, Thompson group F, and kernels of virtual cactus groups.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.4 rests on Lemma 4.7, whose proof omits three sign cases that are used with negative exponents throughout the critical-pair analysis.","rationale":"The reader identified Lemma 4.7 and its missing sign cases as the most fragile load-bearing premise of Theorem 2.4. That identification is accurate: the proof of confluence in Proposition 4.5 reduces every critical pair to Lemmas 4.8–4.12, and Lemmas 4.9, 4.11, and 4.12 explicitly use Lemma 4.7 for negative exponents. Since Theorem 2.4 is the engine for the normal forms, the word problem, parabolic subgroup injectivity, and the preGarside results, a failure here would be consequential. The concern is therefore real as a rigor gap. On the other hand, the missing cases appear derivable from the positive case via the conjugation identity φ_x^a φ_y φ_x^{-a} = φ_{φ_x^a(y)}, using Condition (c) to control order preservation and Lemma 4.6 for the single negative-power base steps. Thus the gap is likely repairable rather than a counterexample to the framework. The verdict should remain conditional pending that verification, so I recommend UNCHANGED relative to the reader's verdict.","tokens_in":57703,"tokens_out":13955,"duration_ms":154075,"concrete_test":"Write out the three omitted sign cases of Lemma 4.7. One route: first prove by induction from Condition (g) that φ_x^m φ_y φ_x^{-m} = φ_{φ_x^m(y)} for every m ∈ Z (the m < 0 step uses Lemma 4.6(2)/(4)), then raise this conjugation identity to the b-th power and verify that z ≤ y implies φ_x^m(z) ≤ φ_x^m(y). If this derivation goes through using only Conditions (a)–(g), the gap is cosmetic; if it requires an additional order-preservation or finiteness hypothesis, Theorem 2.4 needs a revised proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The confluence proof of Theorem 2.4 depends on Lemma 4.7, which states that for z ≤ y ≤ x and all a, b ∈ Z, (φ_x^a ∘ φ_y^b)(z) = (φ_{φ_x^a(y)}^b ∘ φ_x^a)(z). The proof given covers only a > 0, b > 0; the three remaining sign cases are dismissed as 'can be treated in the same way.' These cases are not decorative: Lemma 4.9 Case 2, Lemma 4.11 Case 2, Lemma 4.12 Case 2, and Lemma 7.5 all invoke Lemma 4.7 with negative exponents. If any one of those sign cases failed or required an extra hypothesis absent from Conditions (a)–(g), the resolution of critical pairs in Proposition 4.5 would break, and with it Theorem 2.4, the normal forms, the word problem, and the parabolic subgroup results. As written, this is a real rigor gap at the exact point the framework rests; it is probably fillable by a conjugation argument, but the paper does not supply it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces trickle groups, a class of groups presented by generators with relations of the form xy = zx and x^mu = 1, encoded in a simplicial graph equipped with a partial order, a vertex labeling, and star automorphisms. It claims a terminating and confluent rewriting system on a set of strata, yielding normal forms, a word problem algorithm, a Tits-style algorithm, standard parabolic subgroups with an intersection property, and preGarside/Garside structure in the torsion-free case. Examples include generalized cactus groups, a finite-index subgroup of virtual cactus groups, Thompson group F, and ordered quandle groups.","tokens_in":57986,"tokens_out":5469,"duration_ms":52613,"significance":"If the proof gap identified below is repaired, this is a valuable unifying framework: the rewriting system proof is substantial, and the paper supplies explicit normal forms, a solvable word problem, parabolic subgroup results, and a characterization of when the preGarside monoid is Garside. The new examples, especially Thompson group F and the virtual cactus subgroup, are interesting. The main theorems are proved with considerable detail and the overall architecture is sound, but the current manuscript is not fully verified because the missing sign cases of Lemma 4.7 underpin the confluence proof and all downstream results.","major_comments":[{"comment":"The proof of Lemma 4.7 treats only the case a > 0, b > 0 and dismisses the three remaining sign cases with the phrase \"can be treated in the same way.\" These cases are not decorative: Lemma 4.9 Case 2, Lemma 4.11 Case 2, Lemma 4.12 Case 2, and Lemma 7.5 invoke Lemma 4.7 with negative exponents. Since those lemmas are used to resolve the critical pairs in Proposition 4.5, the confluence of the rewriting system R, and hence Theorem 2.4, Corollary 2.5, Theorem 2.8, and the parabolic subgroup results, rest on this omitted verification. The authors should supply the full four-case proof, or a uniform argument covering all signs, before the main claims can be considered established.","section":"Section 4, Lemma 4.7"},{"comment":"The proof of the claim in Lemma 7.5 applies Lemma 4.7 repeatedly with negative exponents, for example when a syllable of positive exponent c is added and the maps phi_y^{-c} appear. Consequently Theorem 2.15 (the Garside classification) inherits the incompleteness of Lemma 4.7. The authors should either complete Lemma 4.7 for all signs or restructure the argument to avoid reliance on the unproved sign cases.","section":"Section 7, Lemma 7.5"}],"minor_comments":[{"comment":"The proof is said to be identical to that of Lemma 2.1 and is left to the reader, but Lemma 2.1 treats a special case; a full proof, or at least a detailed sketch, is needed to justify the generalized cactus group examples.","section":"Section 3.1, Lemma 3.1"},{"comment":"The three properties of the homeomorphisms h_x are left to the reader; since the identification of Thompson group F as a trickle group is a headline result, this verification should be included or summarized.","section":"Section 3.3, Lemma 3.10"},{"comment":"The proof is left to the reader; it is a short verification, but it should be stated for completeness given that it supports the cactus group parabolic subgroup example.","section":"Section 2.4, Lemma 2.11"},{"comment":"There is a typo: \"Le Υ be a Coxeter graph\" should read \"Let Υ be a Coxeter graph.\"","section":"Section 2, Example 2"},{"comment":"The proof asserts that for non-adjacent x, y the words u^n with u = ({x}, {y}) are R-irreducible; this deserves a one-line justification, for example that no R-transformation applies to ({x}, {y}) because {x, y} is not an edge.","section":"Section 2.2, Corollary 2.7"},{"comment":"The notation eΓ and eTr may be confused with the empty word or with duality; consider a clearer notation. Also, \"preGarisde\" appears in place of \"preGarside\" at several points in the text.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a group theory journal and the framework is promising. The main obstacle is the incomplete proof of Lemma 4.7; I expect the gap is fillable, but it must be addressed before acceptance. I found no issues of circularity or inappropriate citation patterns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the trickle groups paper. The headline: this is a genuinely new unifying framework, and most of the heavy machinery holds up, but the confluence proof contains a real omission that needs fixing before I'd trust Theorem 2.4 as written.\n\nWhat's new: trickle groups unify RAAGs, RACGs, graph products of cyclic groups, and cactus groups under one presentation scheme. The main theorems—normal forms, word problem, parabolic subgroup intersection, preGarside structure—are substantial. The new examples (Thompson F, KVJn, ordered quandle groups) are credible and well-integrated. The paper is honest about what's borrowed: it credits Wyk, Hermiller–Meier, Genevois for the rewriting systems in special cases and Green/Tits for the algorithm form. No fitted parameters; the proofs are mostly direct from the axioms.\n\nThe soft spot is Lemma 4.7. It proves the commutation identity (φ_x^a ∘ φ_y^b)(z) = (φ_{φ_x^a(y)}^b ∘ φ_x^a)(z) only for a>0, b>0 and says the other three sign cases are similar. The stress-test is right that this is load-bearing: Lemmas 4.9, 4.11, 4.12, and 7.5 invoke it with negative exponents. This is a genuine gap in the written proof, not a cosmetic one. I think it's fixable—the dual trickle graph (Example 3) likely handles a<0, and a conjugation/inversion argument handles b<0—but the authors need to supply those cases. Several smaller lemmas (3.1, 3.10, 2.11) are left to the reader; they look true but should be checked.\n\nOverall: the central argument is sound and the framework deserves a serious referee. I'd recommend acceptance conditional on filling Lemma 4.7 and adding details for the left-to-the-reader lemmas. Probably worth a reading group slot in a combinatorial/geometric group theory seminar.","headline":"A genuinely new unifying framework with substantial theorems, but the confluence proof has a fillable yet load-bearing gap in Lemma 4.7 that should be fixed before acceptance.","tokens_in":58492,"tokens_out":2941,"would_cite":true,"duration_ms":31411,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F10","20F05","20F36","20F55","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Trickle groups unify right-angled Artin and Coxeter groups, cactus groups and group F, and give every member a terminating, confluent rewriting system that solves the word problem.","keywords":["trickle groups","right-angled Artin groups","right-angled Coxeter groups","cactus groups","virtual cactus groups","Thompson group F","word problem","preGarside groups"],"falsifier":"Find a trickle graph and vertices $z<y<x$ together with integers $a,b$ of mixed sign for which $(\\varphi_x^a\\circ\\varphi_y^b)(z)\\neq(\\varphi_{\\varphi_x^a(y)}^b\\circ\\varphi_x^a)(z)$. The paper's Lemma 4.6 lists the four sign cases for exponents $\\pm1$; checking those identities in a cactus group built from a finite Coxeter system would settle whether confluence holds in general.","tokens_in":57528,"feed_emoji":"💧","tokens_out":11811,"duration_ms":104016,"temperature":0.7,"pith_summary":"This paper introduces trickle groups, a family of groups defined by presentations whose relations are all of the form $xy=zx$ and $x^\\mu=1$, encoded in a simplicial graph equipped with a partial order on its vertices, a vertex labeling, and automorphisms of the stars of its vertices. The family is broad: it contains right-angled Artin groups, right-angled Coxeter groups, graph products of cyclic groups, cactus groups, the group $F$ of piecewise-linear homeomorphisms, and ordered quandle groups. The central technical result, Theorem 2.4, asserts that for every trickle graph the rewriting system on the set of strata is terminating and confluent, so every element has a unique normal form and, when the vertex set is finite, the word problem is algorithmically solvable. From this the paper derives a Tits-style alternative algorithm, a theory of standard parabolic subgroups whose intersections are again standard parabolic subgroups, and a Garside-theoretic layer: with all labels infinite the associated monoid is preGarside, torsion-free in the finite case, and Garside exactly when the graph is finite and complete.","feed_headline":"Trickle groups: one rewrite system, many word problems","feed_subtitle":"A graph-built family covers Artin, Coxeter, cactus, and group F—each with normal forms.","key_machinery":"The load-bearing object is the set of strata: finite subsets of syllables $x^a$ with $a\\in\\mathbb{Z}_{\\mu(x)}\\setminus\\{0\\}$ and pairwise adjacent supports. Words over this alphabet are pilings, and the rewriting system $R$ consists of $T$-transformations that move a syllable from one stratum to the previous one while applying the star automorphisms $\\varphi_x$, together with a rule deleting empty strata. Confluence is driven by condition (g) in the definition of a trickle graph, the commutation identity $(\\varphi_x^a\\circ\\varphi_y^b)(z)=(\\varphi_{\\varphi_x^a(y)}^b\\circ\\varphi_x^a)(z)$ for $z\\le y\\le x$, which Lemma 4.7 extends to all integer powers $a,b$. The same strata also give the monoid presentation used for the preGarside results.","core_discovery":"The paper's central claim is that trickle groups — groups presented with relations of the form $xy=zx$ and $x^\\mu=1$ governed by a trickle graph — form a single class that inherits the algorithmic and structural good behaviour of the families it generalizes. Concretely, Theorem 2.4 says that the rewriting system $R$ on the set of strata of any trickle graph is a rewriting system for the group, is terminating, and is confluent; Corollary 2.5 then yields a set of normal forms and, for finite vertex sets, a solution to the word problem. The same machinery shows that standard parabolic subgroups are exactly the trickle groups of parabolic subgraphs, that the intersection of two standard parabolic subgroups is again a standard parabolic subgroup, and that the preGarside version (all labels infinite) is a preGarside monoid and group, with the Garside property precisely when the graph is finite and complete.","pith_inferences":["An implicit direction the paper raises but does not settle is whether the normal forms form a regular language and whether trickle groups are automatic or bi-automatic; the terminating and confluent rewriting system makes these questions concrete.","If condition (g) is checked in mixed-sign cases for concrete examples such as dual cactus groups, the word problem algorithm extends to those groups; if one sign case fails, the normal-form theory would need a modified confluence condition.","The Garside characterisation suggests that finite complete preGarside trickle graphs are a source of new Garside groups with explicit Coxeter-style quotients, a direction the paper notes but does not develop.","The same construction that makes virtual cactus groups a trickle group semidirect product of a symmetric group could be applied to other diagram monoids with virtual crossings, though the paper does not explore those."],"forward_implications":["Every trickle group with a finite vertex set has a solvable word problem, by computing the unique $R$-irreducible normal form of any word.","A trickle group is finite exactly when its vertex set is finite, its graph is complete, and all vertex labels are finite.","Standard parabolic subgroups are themselves trickle groups associated with parabolic subgraphs, and the intersection of two standard parabolic subgroups is again a standard parabolic subgroup.","The preGarside trickle monoid embeds into its enveloping group; when the vertex set is finite the enveloping group is torsion-free, and it is a Garside group precisely when the graph is finite and complete.","For virtual cactus groups, the kernel subgroup is a trickle group, which gives a word problem for the virtual cactus group and reproves that the cactus group embeds into it."],"supporting_citations":[{"why":"It supplies the classical theorem that a terminating and confluent rewriting system yields unique normal forms, which is the engine behind Theorem 2.4 and Corollary 2.5.","marker":"[New42]"},{"why":"It provides the model for the Tits-style word problem algorithm that the paper adapts to trickle groups.","marker":"[Tit69]"},{"why":"It provides the analogous word problem algorithm for graph products of cyclic groups, which trickle groups generalize.","marker":"[Gre90]"},{"why":"It supplies the definitions of preGarside monoids and groups and the questions that Theorems 2.14, 2.16, 2.17 and 2.18 answer.","marker":"[GP13]"},{"why":"It supplies the criterion involving short complemented presentations and the sharp $\\theta$-cube condition used to prove that preGarside trickle monoids are preGarside.","marker":"[DDG +15]"},{"why":"It is used in the torsion-free proof when a trickle group splits as an amalgam or a semidirect product.","marker":"[Ser77]"},{"why":"It defines virtual cactus groups and states the embedding result that the trickle presentation of the kernel subgroup reproves.","marker":"[IKL +23]"},{"why":"It provides the basic facts about longest elements of finite Coxeter groups used to build cactus-group trickle graphs.","marker":"[Bou68]"},{"why":"It supplies standard facts about the group $F$, including the generating set used for its trickle presentation.","marker":"[CFP96]"}],"fun_headline_variants":["A single rewrite system tames word problems across Artin, Coxeter, cactus, and F","Trickle groups unify Artin, Coxeter, cactus, and F with one rewrite rule","One rewrite system for many word problems: meet trickle groups","From cactus to Thompson's F: trickle groups cover them all","Trickle groups: a common rewrite rule for diverse group families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a commutation rule among the automorphisms attached to vertices: applying them in different orders must give the same vertex whenever one vertex lies below another, and this must hold for all integer powers, not just positive ones; the paper proves the positive case directly and leaves the three negative sign cases to a similar argument, so a hidden failure there would undo the normal forms and the word problem.","fun_headline_variants_meta":{"raw":{"variants":["A single rewrite system tames word problems across Artin, Coxeter, cactus, and F","Trickle groups unify Artin, Coxeter, cactus, and F with one rewrite rule","One rewrite system for many word problems: meet trickle groups","From cactus to Thompson's F: trickle groups cover them all","Trickle groups: a common rewrite rule for diverse group families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000838,"raw_usage":{"total_tokens":3706,"prompt_tokens":1047,"completion_tokens":2659,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":2556}},"tokens_in":663,"tokens_out":2659,"duration_ms":23676,"temperature":1.0,"reasoning_tokens":2556,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:08:06.938900+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a trickle graph and vertices $z<y<x$ together with integers $a,b$ of mixed sign for which $(\\varphi_x^a\\circ\\varphi_y^b)(z)\\neq(\\varphi_{\\varphi_x^a(y)}^b\\circ\\varphi_x^a)(z)$. The paper's Lemma 4.6 lists the four sign cases for exponents $\\pm1$; checking those identities in a cactus group built from a finite Coxeter system would settle whether confluence holds in general.","supporting_citations":[],"review_version":1}