Spaces of left-preorders on free products

Iv\'an Ch\'ercoles Cuesta

REVIEW 4 major objections 4 minor 21 references

Spaces of left-preorders on free products

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.12676 v1 pith:3YZ44Y3H submitted 2025-07-16 math.GR

Iván Chércoles Cuesta This is my paper · ORCID

classification math.GR MSC 20F6020E0620E0806F15

keywords left-preordersfreeproductsspaceofCantorsetKuroshrankleft-relativelyconvexsubgroupsdynamicalrealizationsBass-Serretheory

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $C$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

There 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.

What 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.

Who 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section 3, proof of Theorem 3.5, Eq. (13) and Eq. (17)] 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 3, proof of Theorem 3.5, Lemma 1.8 applicability] 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 4, Theorem 4.8, Case III] 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 3, proof of Theorem 3.8, infinite Kurosh rank case] 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.

minor comments (4)

[Section 4, definition of K] 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 2, Proposition 2.10] 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.
[Throughout] 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 3, after Eq. (9)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the paper's derivation is self-contained, with auxiliary facts proved in Section 4 and external standard theorems as the only imported inputs.

full rationale

The central derivation chain is not circular. Theorem 3.4 is proved assuming Theorem 3.5, and Theorem 3.5 is proved using dynamical arguments together with Facts I and II, both of which are proven later in Section 4 via Bass-Serre theory, Kurosh's theorem, and the Serre tree construction. Theorem 3.8 then splits into the finite Kurosh rank case (handled by Theorem 3.4) and the infinite Kurosh rank case, which uses the free-factor decomposition of C and the external Proposition 1.2 (from Antolín–Dicks–Šunić) together with Proposition 1.10, whose proof is explicit and does not presuppose the target result. The only self-citation, reference [6], is used solely to fix notation for left-preorders and is not load-bearing for any theorem. Proposition 2.10, used to build the approximating left-preorder, is proved directly via compactness of {+-1,0}^G and the semigroup cone construction, not by assuming the existence of the desired preorder. The reader's identified concern about Lemma 1.9 being applied to words in a larger ball is a possible correctness gap in Equation (13), but it is not a circularity: the claimed conclusion is not equivalent to an input or to a fitted parameter, and no self-referential uniqueness or ansatz is invoked. Therefore the paper is not circular, and the correct circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The proofs rely on standard group theory and two imported dynamical results; no circular fitting.

assumptions (5)

standard math Kurosh subgroup theorem for free products (Theorem 3.1)
Used to define Kurosh rank and to decompose subgroups; cited to [19].
standard math Bass-Serre theory: graph of groups, fundamental group, universal cover, stabilizers (Section 4)
Used to prove Fact I and Fact II; cited to [19].
domain assumption Proposition 1.5: dynamical realization of countable left-preorders by actions on R
Imported from [2]; connects preorders to Homeo+(R) actions.
domain assumption Proposition 1.2: factors of free products are left-relatively convex (from [3])
Used to extend preorders from C to C1 in Theorem 3.8.
standard math Tychonoff theorem and Cantor space characterization (Prop 2.7)
Used to prove compactness and to identify Cantor sets.

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Cite this review

Pith. "Pith review of Spaces of left-preorders on free products." pith.science (2026). https://pith.science/paper/3YZ44Y3H

@misc{pith2026250712676,
  author       = {Pith},
  title        = {Pith review of: Spaces of left-preorders on free products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3YZ44Y3H}},
  note         = {Machine review of arXiv:2507.12676}
}

read the original abstract

Using dynamical techniques we show that there are no isolated elements on the space of left-preorders on a free product of two groups. As a consequence, when the groups are finitely generated, this space is either empty or a Cantor set. For any subgroup of a free product having finite Kurosh rank, we develop similar results for the subspace consisting on the set of left-preorders relative to that subgroup. We provide a generating set for a certain intersection of subgroups of a free product.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

& Schwabrow, I

Antol ´ ın, Y., Martino, A. & Schwabrow, I. Kurosh rank of intersections of subgroups of free products of right-orderable groups. Math. Res. Lett. . 21, 649-661 (2014), https://doi.org/10.4310/MRL.2014.v21.n4.a2

work page doi:10.4310/mrl.2014.v21.n4.a2 2014

[2] [2]

& Rivas, C

Antol ´ ın, Y. & Rivas, C. The space of relative orders and a generalization of Morris indicability theorem. J. Topol. Anal.. 13, 75-85 (2021), https://doi.org/10.1142/S1793525320500053

work page doi:10.1142/s1793525320500053 2021

[3] [3]

&ˇSuni´ c, Z

Antol ´ ın, Y., Dicks, W. &ˇSuni´ c, Z. Left relatively convex subgroups.Topological Methods In Group Theory. 451 pp. 1-18 (2018)

work page 2018

[4] [4]

& Triestino, M

Brum, J., Bon, N., Rivas, C. & Triestino, M. Locally moving groups and laminar actions on the line. (2024), https://arxiv.org/abs/2104.14678

arXiv 2024

[5] [5]

& Kam, S

Burns, R., Chau, T. & Kam, S. On the rank of intersection of subgroups of a free product of groups. J. Pure Appl. Algebra. 124, 31-45 (1998), https://doi.org/10.1016/S0022-4049(98)00120-0

work page doi:10.1016/s0022-4049(98)00120-0 1998

[6] [6]

The space of Conradian left-preorders

Ch´ ercoles Cuesta, I. The space of conradian left-preorders. (2025), https://arxiv.org/abs/2506.18573

work page Pith review arXiv 2025

[7] [7]

& Rond, G

Decaup, J. & Rond, G. Preordered groups and valued fields. (2019), https://arxiv.org/abs/1912.03928

arXiv 2019

[8] [8]

& Rivas, C

Deroin, B., Navas, A. & Rivas, C. Groups, Orders, and Dynamics. (2016), https://arxiv.org/abs/1408.5805

arXiv 2016