REVIEW 1 major objections 4 minor 35 references
Minimal sofic shift on a group that is not finitely-generated
T0 review · 1 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper constructs a non-finitely-generated group $(F_4 \times F_2) \rtimes F_\infty$ that carries an infinite minimal sofic shift, answering a question of Doucha, Melleray and Tsankov.
desk verdict A real result: first non-finitely-generated group with a minimal sofic shift; proof likely correct but Lemma 16 hides a load-bearing combinatorial step that needs to be spelled out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the combination of antidiagonal minimality with minimal $\Phi$-joinings. An action is antidiagonally minimal when its diagonal action on every finite tuple of distinct points is minimal; Thompson's $V$ acting on Cantor space has this property, and since $V$ is $3/2$-generated, a copy of $F_2$ can act this way. Minimal $\Phi$-joinings is a disjointness condition saying that the base system is disjoint from all its conjugates by a chosen family of automorphisms $\Phi$; Lemma 15 shows that this condition exactly characterizes when inducing the base action to a semidirect product preserves minimality of the normal-subgroup action. Lemma 16 then uses antidiagonal minimality to separate coincident coordinates while avoiding new coincidences, and Lemma 17 wraps the argument with the simulation and self-simulability theorems that make the resulting system sofic.
What would settle it
Find a concrete instance of Lemma 16 where the separation claim fails: a finite tuple with one equal pair, a B-element, and prescribed clopen sets such that every A-element that puts the equal pair into the right part of the commutator's support also forces some previously distinct pair to collide after applying B. Since elements of Thompson's V are prefix-code permutations, such an instance could in principle be found by a finite search on small tuples; exhibiting one would break the minimality proof, even if the theorem itself might survive through another route.
Extended reading notes
Core claim
The central discovery is that the obstruction to minimal sofic shifts on non-finitely-generated groups is not finite generation itself but the extra structure of the group. The paper constructs a sofic subshift on $G = (F_4 \times F_2) \rtimes F_\infty$ with no proper nonempty closed $G$-invariant subset. Concretely, one first makes a sofic shift on $F_4 \times F_2$ where the second free factor acts trivially; inducing this shift to a semidirect product $(F_4 \times F_2) \rtimes F_2$ makes the $F_4$-subaction minimal, and this minimality survives passage to the subgroup $(F_4 \times F_2) \rtimes F_\infty$. The proof combines Thompson's $V$ as a source of very high transitivity with simulation theorems that turn computable actions into sofic shifts.
Load-bearing premise
The proof depends on the assertion in Lemma 16 that one element of the antidiagonally minimal group can be chosen to move every coordinate of a finite tuple into prescribed disjoint clopen sets while ensuring that the subsequent action of a B-element creates no new equal coordinate pairs; the paper says this is clear and sketches the separation rather than proving it in detail.
Editorial extensions
If this is right
- Question 7.18(ii) of Doucha, Melleray and Tsankov has a positive answer: an infinite minimal sofic shift exists on a non-finitely-generated group.
- The example is explicit and belongs to a familiar family: $(F_4 \times F_2) \rtimes F_\infty$, with the $F_\infty$ action given by conjugation inside a copy of $F_2$.
- Any non-finitely generated group carrying such a shift must lie outside the amenable and locally finite obstruction classes; the constructed group does.
- The proof gives a reusable criterion (Lemma 17): from an antidiagonally minimal computable action, an expansive computable action, and a nonamenable or product factor, one obtains a minimal sofic shift on a semidirect product.
- The analogous question for subshifts of finite type (Question 1) remains open, but the sofic case no longer stands in the way.
Reading between the lines
- If the Lemma 16 separation step can be made fully rigorous, the same construction likely works with any group that has an antidiagonally minimal computable action and a centerless quotient, giving many non-finitely-generated examples rather than one.
- Since non-finitely generated groups with projectively isolated sofic shifts must be minimal, this example is a natural test case for whether projectively isolated sofic shifts can be dense, which by the paper's cited equivalence would yield a generic conjugacy class of Cantor actions on some non-finitely generated group.
- One could try to upgrade the simulation step to make the shift an SFT; if that succeeded, Question 1 would also be answered affirmatively, but the current proof does not supply such a cover.
- The reliance on Thompson's $V$ suggests that high transitivity, rather than finite generation, is the dynamical property that enables minimality in this setting; testing other highly transitive groups would clarify the scope.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper answers Question 7.18(ii) of Doucha, Melleray and Tsankov by proving Theorem 1: there exists a group of the form (F4×F2)⋊F∞, not finitely generated, that admits an infinite minimal sofic shift. The method combines known self-simulability results for direct products of nonamenable groups (Barbieri–Sablik–Salo and Barbieri) with the antidiagonal minimality of Thompson's V. Section 7 contains the core Lemma 16, which gives sufficient conditions for the A∗B-subaction of the induced system X^D to be minimal; Theorem 5 and Corollary 1 instantiate the lemma with A=B=C=F2 and a suitable copy D≅F∞ inside V. The final step identifies the minimal system with a sofic shift on (F4×F2)⋊F∞.
Significance. If the proof is completed as sketched, the result answers in the negative the open question whether only finitely generated groups admit nontrivial minimal sofic shifts, at least outside the amenable and locally finite obstruction. The construction is elegant and rests on independent published theorems rather than on the desired conclusion, so I see no circularity. The paper's main contribution is the combination of antidiagonal minimality of Thompson's V with self-simulability, packaged as a reusable criterion in Lemma 16. The remaining gap in the proof of Lemma 16 is localized but load-bearing.
major comments (1)
- [Lemma 16 (Section 7)] The separation step in Lemma 16 is under-proved and is load-bearing for Theorem 5 and Theorem 1. After b∈B is chosen so that [ϕ_{f′f^{-1}}, b] has nonempty support, the proof must produce a∈A such that (i) each equality class of the finite tuple is mapped to the prescribed open subset, so that the chosen pair separates, and (ii) for every previously distinct pair of coordinates (p,q), the values u_p,u_q chosen for the distinct class representatives avoid the relation u_q = (ϕ_{q^{-1}}bϕ_q)^{-1}(ϕ_{p^{-1}}bϕ_p)u_p, so that the B-twist does not create a new coincidence. The text asserts existence of such a with 'It is clear...' and gives no proof. One also needs to know that b can be chosen so that none of the finitely many relation maps is the identity; this is true for Thompson's V because it is infinite simple and hence not a union of finitely many proper centralizers, but this is not stated. Once this single-step statement is proved, iterating over equality classes is immediate because group elements act by homeomorphisms of X^F, so separated pairs remain separated, but that iteration is also not written out.
minor comments (4)
- [Lemma 16, proof] The expression 'for b ∈ b' should read 'for b ∈ B', and 'outside the center of the computable quotient of the B-action' should read 'outside the center of the faithful quotient G'.
- [Corollary 1] The proof should explicitly identify the shift on (F4×F2)⋊F∞ as the induction of the sofic shift from Theorem 5 to this subgroup, and should cite or prove the standard fact that induction preserves soficness; the minimality of the full action follows from minimality of the F4-subaction, but this implication is also not stated.
- [Theorem 5, item 3] The action of B=F2 on X is via a fixed surjection F2→V; the wording 'also B to act by the natural action of V' should name this surjection to make the faithful quotient G=V unambiguous.
- [Section 1.2] The statement that sofic shifts arise from induction from a finitely generated subgroup because their SFT covers do is implicitly used later; a reference or a one-sentence proof would help the reader.
Circularity Check
No significant circularity: the derivation is a genuine application of external simulation theorems and properties of Thompson's V.
full rationale
The derivation chain is self-contained relative to the cited external theorems and contains no fitted parameters or predictions that reduce to their inputs by construction. The main result combines: (i) the antidiagonal minimality of Thompson's V (Lemma 7); (ii) Lemma 16, which transfers this property to the A*B-subaction of an induced system; and (iii) the self-simulability theorems Lemma 1 (from [8]) and Lemma 2 (from [5]), which make the constructed system sofic. The cited results co-authored by Salo, namely the self-simulability theorem of [8] and the terminology reference [9], are not load-bearing in a circular sense: [8] is an independently published theorem with a proof whose assumptions do not include the target result, and [9] is cited only for terminology. Definition 4 introduces 'minimal Phi-joinings' precisely as disjointness of the family {X^phi}, and Lemma 15 is a direct coordinate-by-coordinate translation of that definition into the induced action; because the paper openly presents this as a definition and the substantive content lies in showing the property is realizable via Thompson's V, this is not a masked prediction. The flagged weakness in Lemma 16 ('It is clear that by choosing a suitably...') is an omitted proof of the separation claim needed for minimality; it is a correctness or rigor gap, not a circular step, since it does not presuppose the conclusion. Corollary 1 also leaves implicit the standard fact that induction preserves soficity; again this is a missing justification, not circularity. Overall, the central claim does not reduce to its own inputs by construction, and no circular step is exhibited.
Assumptions & free parameters
assumptions (4)
- domain assumption Self-simulability of direct products of nonamenable finitely generated groups (Lemma 1, from Barbieri-Sablik-Salo).
- domain assumption Pullback of every computable G-subshift to GxHxK is sofic (Lemma 2, from Barbieri).
- domain assumption Thompson's V is 3/2-generated (Lemma 12, from Donoven and Harper); V contains a free subgroup (Lemma 11, proved in the paper).
- standard math F2 contains a copy of F_infinity (standard free group property).
Cite this review
Pith. "Pith review of Minimal sofic shift on a group that is not finitely-generated." pith.science (2026). https://pith.science/paper/RURZU7TG
@misc{pith2026250706599,
author = {Pith},
title = {Pith review of: Minimal sofic shift on a group that is not finitely-generated},
year = {2026},
howpublished = {\url{https://pith.science/paper/RURZU7TG}},
note = {Machine review of arXiv:2507.06599}
}
abstract
We prove that there exists a group which is not finitely generated, but admits a minimal sofic shift. This answers a question of Doucha, Melleray and Tsankov. The group is of the form $(F_4 \times F_2) \rtimes F_{\infty}$. The construction itself is based on simulation theory and properties of Thompson's~$V$.
Reference graph
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