{"id":"904c0880-e182-45bf-9028-bef0c7bf28b4","arxiv_id":"2507.12676","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Left-preorder spaces of free products have no isolated elements; when the factors are finitely generated, every nonempty such space is a Cantor set.","lead":"This paper proves that the space of left-preorders on a free product of two groups has no isolated points, and is a Cantor set when the factors are finitely generated. It also provides a new theorem describing certain intersections of subgroups of free products by explicit generating sets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (13) in Theorem 3.5 is derived by applying Lemma 1.9 with extrema from B_{X_G}(n) to a ball B_{X_G∪X_H}(n+1); H-letter prefixes can exceed the interval, so the equality and hence the central theorem are unsupported.","rationale":"The reader's weakest assumption correctly identifies the point where the proof of Theorem 3.5 is least secure, and I agree with that assessment. The issue is not merely a missing detail: the proof of Eq. (17) explicitly asserts h lambda_n^+ C_Gamma ⪯ lambda_{n+1}^+ C_Gamma from the fact that h lambda_n^+ belongs to B_{X_G∪X_H}(n+1), but lambda_{n+1}^+ is defined as maximal in B_{X_G}(n+1). There is no reason an H-letter prefix is bounded above by an X_G-maximal element; the lexicographic example shows the bound can fail. This is the same confusion the reader flagged in the derivation of Eq. (13), and it is load-bearing: Eq. (13) is used to show the new preorder agrees with the original on T and to establish Eq. (14). If Eq. (13) fails, the constructed preorder may not have the required properties. I do not see a way to rescue the written argument without changing the construction, for example by taking maximal and minimal elements in the full ball B_{X_G∪X_H}(n), which would require rechecking Lemma 1.8 and Fact II. The Theorem 4.8 Case III concern about membership in J is also real, but the Eq. (13) gap is enough to block the central claim. Since the theorem may be repairable and no formal verification or independent implementation supports the proof, a conditional verdict is appropriate; the reader's verdict should stand.","tokens_in":33654,"tokens_out":10923,"duration_ms":122582,"concrete_test":"Check whether Eq. (17) can be repaired by choosing lambda_n^+ and lambda_n^- as maximal/minimal in the full ball B_{X_G∪X_H}(n) rather than B_{X_G}(n), and verify that Lemma 1.8 and Fact II still apply to that choice. In parallel, run a concrete instance: G = H = Z, C = {1}, X_G = {a^{±1}}, X_H = {b^{±1}}, lexicographic order on F(a,b) with b highly positive; for n = 1 compute D(h lambda_n^+)(0) and D(lambda_{n+1}^+)(0). If D(ba)(0) > D(a^2)(0), the displayed inequality used to prove Eq. (17) fails, confirming the misapplication of Lemma 1.9.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main construction in Theorem 3.5 needs Eq. (13): D_phi(w)(0) = D(w)(0) for every w in B_{X_G∪X_H}(n+1). To obtain it, the proof invokes Lemma 1.9(2), whose interval [D(lambda_n^-)(0), D(lambda_n^+)(0)] is built from lambda_n^+/- being maximal/minimal in B_{X_G}(n), not in B_{X_G∪X_H}(n). Lemma 1.9(1) guarantees only that prefixes formed from X_G lie in this interval; prefixes containing H letters have no such bound. The paper's Eq. (17) tries to repair this by proving D_phi(h)(x) = D(h)(x) for x <= D(lambda_n^+)(0), but its key inequality is itself invalid: it compares h lambda_n^+ in B_{X_G∪X_H}(n+1) with lambda_{n+1}^+, and lambda_{n+1}^+ is maximal only among X_G-words of length at most n+1. A word h lambda_n^+ using H can be larger. For example, in F(a,b) with a lexicographic order in which b is very positive, take X_G = {a^{±1}} and X_H = {b^{±1}}; then h lambda_n^+ = b a^n is larger than lambda_{n+1}^+ = a^{n+1}, contradicting the displayed inequality. Consequently D_phi and D need not agree on the claimed interval, and Eq. (13) does not follow from Lemma 1.9. Since Eq. (13) is used to prove both C_Gamma ≺_* t C_Gamma and Eq. (14), the proof of Theorem 3.5, and with it Theorems 3.4 and 3.8, is not established as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the space PO(G*H) of non-trivial left-preorders on a free product of two groups. The main theorem (Theorem 3.8 and Corollary 3.10) asserts that if G and H are non-trivial, then PO(G*H) has no isolated elements; if G and H are finitely generated left-orderable groups, PO(G*H) is homeomorphic to the Cantor set. The proof passes through a relative version PO_C(G*H) for subgroups C of finite Kurosh rank (Theorems 3.4 and 3.5), using dynamical realizations on the line, a perturbation by a homeomorphism supported away from the reference point, and a Bass-Serre description of intersections (Theorem 4.8). The paper also derives consequences about positive cones not being finitely generated as semigroups.","tokens_in":34060,"tokens_out":8211,"duration_ms":91705,"significance":"If correct, the main results give a clean dichotomy for the space of left-preorders on free products and extend Rivas's theorem for left-orders to left-preorders, including a Cantor-set conclusion under finite generation. The auxiliary Theorem 4.8, giving a uniform finite generating set for intersections C ∩ ⟨G0,H0⟩, is potentially independently useful. The paper is self-contained, uses standard tools (Kurosh's theorem, Bass-Serre theory, Tychonoff compactness) in a direct way, and does not rely on fitted parameters or post-hoc assumptions. However, several load-bearing points in the central proof are not justified as written.","major_comments":[{"comment":"The derivation of Eq. (13) is not justified. The proof of Eq. (17) asserts that for h ∈ XH and x ≤ D(λ_n^+)(0), the inequality hλ_n^+ CΓ ⪯ λ_{n+1}^+ CΓ holds, but λ_{n+1}^+ is maximal only in B_{XG}(n+1), while hλ_n^+ lies in B_{XG∪XH}(n+1) and may involve an H-letter. Nothing in the setup prevents hλ_n^+ CΓ from being larger than λ_{n+1}^+ CΓ; for example, in a free group with a lexicographic order where a generator of H is very positive, h a^n can exceed a^{n+1}. Consequently the inequality D(hλ_n^+)(0) ≤ D(λ_{n+1}^+)(0) is unsupported, and Lemma 1.9(2) cannot be applied with the stated interval to conclude Eq. (13). Since Eq. (13) is used to prove both the equality for D_phi(gλ_{n+1}^+) and the equality for D_phi(t)(0), the proof of Theorem 3.5, and with it Theorems 3.4 and 3.8, is not established as written.","section":"Section 3, proof of Theorem 3.5, Eq. (13) and Eq. (17)"},{"comment":"The proof applies Lemma 1.8 to the sequence λ_n^+ of maximal elements in B_{XG}(n) and concludes that λ_n^+ C ≠ λ_m^+ C for n ≠ m. Lemma 1.8 requires C ∩ ⟨XG⟩ ≠ ⟨XG⟩, i.e. the existence of y ∈ XG with C ≺ yC. The text only requires that XG be a finite symmetric subset of G with at least one non-trivial element and with T ∪ Ω ⊆ ⟨XG⟩ * ⟨XH⟩. It does not ensure that any element of XG is ⪯-positive relative to C. If C contains G, then every XG ⊆ G is contained in C, and λ_n^+ C = C for all n, so Fact II cannot be applied. This is a load-bearing gap in the construction of γ*.","section":"Section 3, proof of Theorem 3.5, Lemma 1.8 applicability"},{"comment":"In Case III of the induction, the proof asserts that a vertex v'_n = yv_G belonging to VT0 is in J, citing the definition of J. But J consists of vertices xv_G with Stab_C(xv_G) ≠ {1}. Membership in T0 or in the fundamental core of TC does not by itself imply a non-trivial C-stabilizer; the fundamental core also contains vertices lying on reduced closed paths based at u*. No argument is given that the particular vertex at which the path leaves T0 has non-trivial stabilizer. This affects the definition of G_{v'_n} and the element α_{v'_n} used to prove that Vgg^{-1} is in ⟨S⟩. The same issue appears in Case IV. The proof of Theorem 4.8 is therefore incomplete in these cases.","section":"Section 4, Theorem 4.8, Case III"},{"comment":"In the infinite Kurosh rank case of Theorem 3.8, the proof chooses a proper free factor C1 of C containing R ∩ C and writes C = C1 * C2. It then claims that C1 is left-relatively convex on C by Proposition 1.2. Proposition 1.2 applies to free products of non-trivial left-orderable factors, but C2 is not known to be left-orderable, and the groups G and H are not assumed left-orderable in Theorem 3.8. Thus the cited proposition does not justify the relative convexity of C1 in C as used to apply Proposition 1.10. The infinite-rank case of Theorem 3.8 is therefore not supported by the given argument.","section":"Section 3, proof of Theorem 3.8, infinite Kurosh rank case"}],"minor_comments":[{"comment":"In the definition of K, the condition is written as Stab_C(xv_G) ≠ {1}, but it should be Stab_C(xv_H) ≠ {1} to match the H-vertex type.","section":"Section 4, definition of K"},{"comment":"In the definition of \\tilde{A}_{{f1,...,fl}}, the last displayed set uses V^1_{f_n} while the preceding notation uses f_l; this is a typo that makes the formula inconsistent.","section":"Section 2, Proposition 2.10"},{"comment":"There are many typographical errors that should be corrected: 'defintion', 'done in done in', 'follwos', 'elemnts', 'sefl-contained', 'tranvsersal', and missing spaces such as 'POC(G)' and 'PO C(G)'.","section":"Throughout"},{"comment":"The displayed chain after Eq. (9) reads 'λ_{n+1}^+C ≺ hλ_{n+1}^+ ≺ gλ_{n+1}^+C'; the middle term should be hλ_{n+1}^+C for consistency of coset notation.","section":"Section 3, after Eq. (9)"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Eq. (13) is real and, on reading the paper, it does land: the proof of Theorem 3.5 has a genuine gap in the application of Lemma 1.9. I also found additional load-bearing issues in the applicability of Lemma 1.8 and in the use of Proposition 1.2 in the infinite Kurosh rank case. I do not recommend rejection because the main claims may be salvageable with a substantial reworking of the perturbation argument and the auxiliary subgroup lemma, but the current version is not a sound proof of the central theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plain take: this is a serious extension of Rivas' dynamical proof to left-preorders on free products, with a genuinely new intersection theorem (4.8) that could be useful beyond this paper. The main theorem — PO(G*H) has no isolated elements, Cantor set in the finitely generated case — is the right conjecture and the strategy is sound in outline. But as written, the proof of Theorem 3.5 has a load-bearing gap.\n\nThe problem is Equation (13). The proof applies Lemma 1.9(2) with λ_n^± chosen as extrema in B_XG(n), then concludes equality on B_{XG∪XH}(n+1). Lemma 1.9(2) only gives the conclusion for the ball generated by the same set used to define the extrema, and Lemma 1.9(1) bounds only prefixes built from XG. Words containing H letters are not controlled. The attempted repair in Equation (17) compares hλ_n^+ with λ_{n+1}^+, but λ_{n+1}^+ is maximal only among XG-words; hλ_n^+ can exceed it (e.g., in F(a,b) with b very positive, b a^n ≻ a^{n+1}). So D_phi and D need not agree on the claimed interval, and Eq (13) — used for both CΓ≺∗tCΓ and Eq (14) — does not follow. Theorem 3.5, and with it Theorems 3.4 and 3.8, are not established as written.\n\nThere is also a smaller gap in Theorem 4.8, Case III: the argument asserts v'_n ∈ J, but J requires Stab_C(xv_G)≠{1}, and that is not shown for a vertex along the path from 1v0 to xv0. It may be true that such vertices lie in the core, but core membership alone does not give non-trivial stabilizer. This is repairable but needs a real argument.\n\nWhat is good: the paper is honestly written, cites the relevant literature (Rivas, Muliarchyk, Antolín–Rivas), and the dynamical framework is adapted carefully. Theorem 4.8, if fixed, is a nice standalone result on intersections in free products, going beyond the strengthened Hanna Neumann-type bounds. The topological corollaries (Cantor set, uncountability) are natural and the reduction steps are well explained. No signs of circularity or fitted parameters; the dependence on Kurosh rank and Bass–Serre is legitimate.\n\nWho this is for: geometric group theorists working on orderability spaces. It deserves a serious referee — the claims are important enough and the approach is mostly right — but a referee should be asked to check the proof of Theorem 3.5 line by line, especially the derivation of Eq (13). I would send it to review with a request for major revision rather than desk-reject.","headline":"The main topological claims are plausible but the proof of Theorem 3.5 relies on an invalid application of Lemma 1.9, so the paper needs major repair.","tokens_in":34593,"tokens_out":3740,"would_cite":false,"duration_ms":36878,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F60","20E06","20E08","06F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the space of left-preorders on a free product of two non-trivial groups has no isolated points, so it is either empty or a Cantor set.","keywords":["left-preorders","free products","space of left-preorders","Cantor set","Kurosh rank","left-relatively convex subgroups","dynamical realizations","Bass-Serre theory"],"falsifier":"For the infinite dihedral group $\\mathbb{Z}/2\\mathbb{Z} * \\mathbb{Z}/2\\mathbb{Z}$, enumerate the non-trivial left-preorders; if any is isolated, Theorem 3.8 fails. A second check targets the proof's key step: pick $G=H=\\mathbb{Z}$ with the standard ordering, choose finite $X_G,X_H$, and verify whether all $H$-prefixes of words in $B_{X_G\\cup X_H}(n+1)$ land inside $[D(\\lambda^-_n)(0), D(\\lambda^+_n)(0)]$; if not, Equation (13) does not follow from Lemma 1.9.","tokens_in":33443,"feed_emoji":"📐","tokens_out":7921,"duration_ms":82559,"temperature":0.7,"pith_summary":"Left-preorders on a group are total, transitive, reflexive relations invariant under left multiplication; a left-preorder generalizes a left-order by allowing distinct elements to be equivalent, the equivalence class of the identity being a proper subgroup. The paper proves that for any two non-trivial groups $G$ and $H$, the space $\\mathrm{PO}(G*H)$ of non-trivial left-preorders on the free product $G*H$ has no isolated points. Consequently, for finitely generated left-orderable $G$ and $H$, this space is homeomorphic to the Cantor set. The same no-isolated-points conclusion is proved for the relative spaces $\\mathrm{PO}_C(G*H)$ when $C$ has finite Kurosh rank, and the proof produces a generating set for intersections $C \\cap \\langle G_0,H_0\\rangle$ that is of independent interest.","feed_headline":"No isolated left-preorders on any free product of two groups","feed_subtitle":"When the factors are finitely generated, the space is empty or a Cantor set — nothing in between.","key_machinery":"A left-preorder is encoded by a decomposition $G = P \\sqcup P^{-1} \\sqcup C$, where $P$ is a subsemigroup and $C$ is a proper subgroup; the positive cone $P$ and the subgroup $C$ determine the preorder. The proof works with a dynamical realization $D \\colon G \\to \\mathrm{Homeo}_+(\\mathbb{R})$ whose stabilizer of $0$ is $C$, so comparisons of cosets become comparisons of real numbers. Given a preorder, the argument selects a very positive element $\\lambda^+_{n+1}$ and elements $g \\in X_G$, $h \\in X_H$, then conjugates the $H$-part by an orientation-preserving homeomorphism supported on an interval strictly between $D(h\\lambda^+_{n+1})(0)$ and $D(g\\lambda^+_{n+1})(0)$. The perturbed representation $D_\\phi$ agrees with $D$ on an initial ball, so it induces a new left-preorder that agrees with the original on any prescribed finite set but orders $\\gamma_* = (h\\lambda^+_{n+1})^{-1}g\\lambda^+_{n+1}$ oppositely. The finite-Kurosh-rank control comes from Bass–Serre theory: the fundamental core of the quotient graph is finite, which supplies both the finiteness of the data and the generating set for intersections.","core_discovery":"The central claim is that no left-preorder on a free product of two non-trivial groups sits alone in its space: every basic open neighbourhood of any preorder contains another, different preorder. This is Theorem 3.8. Because the space is compact, Hausdorff, totally disconnected and metrizable whenever the factors are finitely generated, the absence of isolated points forces $\\mathrm{PO}(G*H)$ to be a Cantor set — or empty — when $G$ and $H$ are finitely generated left-orderable groups. The relative version Theorem 3.4 states the same for $\\mathrm{PO}_C(G*H)$ with $C$ of finite Kurosh rank, and Corollary 3.12 concludes that the positive cone of any such left-preorder is not finitely generated as a subsemigroup.","pith_inferences":["The two-factor theorem likely iterates to free products of any finite number on non-trivial factors, since $G_1*(G_2*G_3)$ is again a free product of two non-trivial groups; the paper does not state this extension.","Theorem 4.8 invites sharpness tests: the paper proves a bound on the Kurosh rank of the intersection but does not say whether the bound is attained in natural examples.","The same compactness-plus-perturbation method may apply to more general graphs of groups, where a 'finite Kurosh rank' condition would play the role of the finiteness of the fundamental core; this direction is not pursued here."],"forward_implications":["For non-trivial finitely generated left-orderable $G$ and $H$, the space $\\mathrm{PO}(G*H)$ is homeomorphic to the Cantor set.","For any finitely generated left-relatively convex subgroup $C$ of $G*H$, the relative space $\\mathrm{PO}_C(G*H)$ has no isolated elements, and is a Cantor set when $G$ and $H$ are finitely generated.","No left-preorder relative to a subgroup of finite Kurosh rank has a positive cone finitely generated as a semigroup.","If $\\mathrm{PO}(G*H)$ is non-empty, it is uncountable.","Theorem 4.8 gives explicit finite data controlling the intersection of a finite-Kurosh-rank subgroup $C$ with an arbitrary subgroup $\\langle G_0,H_0\\rangle$, bounding the Kurosh rank of the intersection by that of $C$."],"supporting_citations":[{"why":"Supplies the dynamical strategy, Lemma 1.9 and the prior result for left-orderings of free products that the paper adapts to left-preorders.","marker":"[18]"},{"why":"Establishes the no-isolated-points result for free groups of finite rank that motivates the programme.","marker":"[14]"},{"why":"Provides the compactness and property-($E_C$) framework used in Proposition 2.10 to build left-preorders.","marker":"[2]"},{"why":"Gives the notion of left-relatively convex subgroups and Proposition 1.2 used to extend preorders.","marker":"[3]"},{"why":"Provides Bass–Serre theory, Kurosh's theorem and the graph-of-groups machinery used to establish Facts I and II.","marker":"[19]"},{"why":"Defines Kurosh rank and proves its well-definedness for subgroups of free products.","marker":"[5]"},{"why":"Context for intersections of subgroups of free products of right-orderable groups, generalised by Theorem 4.8.","marker":"[1]"},{"why":"The bi-order analogue for free products, showing the Cantor-set conclusion is part of a family of known results.","marker":"[16]"}],"fun_headline_variants":["No isolated left-preorders on free products","Free product left-preorders: all points cluster","Left-preorder space is Cantor or empty","No standalone left-preorders in free product spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Theorem 3.5 assumes that the interval between the images of the most positive and most negative elements of the $G$-ball $B_{X_G}(n)$ already contains the images of every prefix of every word in the larger ball $B_{X_G \\cup X_H}(n+1)$, including prefixes that use letters of $H$; this containment is not explicitly established.","fun_headline_variants_meta":{"raw":{"variants":["No isolated left-preorders on free products","Free product left-preorders: all points cluster","Left-preorder space is Cantor or empty","No standalone left-preorders in free product spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000143,"raw_usage":{"total_tokens":1088,"prompt_tokens":780,"completion_tokens":308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":250}},"tokens_in":396,"tokens_out":308,"duration_ms":4212,"temperature":1.0,"reasoning_tokens":250,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:44:11.364848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the infinite dihedral group $\\mathbb{Z}/2\\mathbb{Z} * \\mathbb{Z}/2\\mathbb{Z}$, enumerate the non-trivial left-preorders; if any is isolated, Theorem 3.8 fails. A second check targets the proof's key step: pick $G=H=\\mathbb{Z}$ with the standard ordering, choose finite $X_G,X_H$, and verify whether all $H$-prefixes of words in $B_{X_G\\cup X_H}(n+1)$ land inside $[D(\\lambda^-_n)(0), D(\\lambda^+_n)(0)]$; if not, Equation (13) does not follow from Lemma 1.9.","supporting_citations":[{"cited_title":"Left-orderings on free products of groups","cited_arxiv_id":null,"evidence_quote":"Supplies the dynamical strategy, Lemma 1.9 and the prior result for left-orderings of free products that the paper adapts to left-preorders."},{"cited_title":"Free lattice-ordered groups represented as o-2 transitive l-permutation groups","cited_arxiv_id":null,"evidence_quote":"Establishes the no-isolated-points result for free groups of finite rank that motivates the programme."},{"cited_title":"& Rivas, C","cited_arxiv_id":null,"evidence_quote":"Provides the compactness and property-($E_C$) framework used in Proposition 2.10 to build left-preorders."},{"cited_title":"&ˇSuni´ c, Z","cited_arxiv_id":null,"evidence_quote":"Gives the notion of left-relatively convex subgroups and Proposition 1.2 used to extend preorders."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides Bass–Serre theory, Kurosh's theorem and the graph-of-groups machinery used to establish Facts I and II."},{"cited_title":"& Kam, S","cited_arxiv_id":null,"evidence_quote":"Defines Kurosh rank and proves its well-definedness for subgroups of free products."},{"cited_title":"& Schwabrow, I","cited_arxiv_id":null,"evidence_quote":"Context for intersections of subgroups of free products of right-orderable groups, generalised by Theorem 4.8."},{"cited_title":"Free products of bi-orderable groups","cited_arxiv_id":"2403.14779","evidence_quote":"The bi-order analogue for free products, showing the Cantor-set conclusion is part of a family of known results."}],"review_version":1}