{"id":"cc7d22ad-93f5-4dba-a238-e2f4071aed4d","arxiv_id":"2507.06599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The group (F4 x F2) rtimes F_infinity, which is not finitely generated, is shown to admit an infinite minimal sofic shift, answering Question 7.18(ii) of Doucha, Melleray and Tsankov.","lead":"Ville Salo proves that a certain group that is not finitely generated, namely (F4 x F2) rtimes F_infinity, still admits a minimal sofic shift, answering an open question of Doucha, Melleray and Tsankov. The construction combines Thompson's group V, semidirect products and simulation theory, and it opens a path for studying generic dynamics on non-finitely-generated groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 16's 'it is clear' separation step is under-proved: eliminating all equal coordinate pairs requires an induction the paper does not give, and this is the crux of the minimality claim.","rationale":"The central claim (Theorem 1) rests on Lemma 16: the minimality of the induced system is proved by showing that for every finite coordinate set F, the A*B-action on X^F can move any tuple into any open set. The only non-formal step is the separation of equal coordinate pairs. I checked the surrounding algebra: the commutator computation g = phi_{f'^{-1}} b^{-1} phi_{f'} phi_{f^{-1}} b phi_f conjugate to [phi_{f' f^{-1}}, b] is correct, and the existence of b with nontrivial commutator follows from injectivity into G/Z(G) and faithfulness. What is missing is the combinatorial control needed to separate all equal pairs without creating new ones. The paper says 'It is clear that by choosing a suitably...', which is precisely the point where a rigorous proof must handle multiple equality classes, classes of size > 2, and the preservation of previously achieved separations. This is not a fatal flaw: an induction on the number of equal pairs, using finite intersections of nowhere dense bad sets and antidiagonal minimality to place values, should go through. But because the paper does not supply that induction, the proof is not fully checkable as written; a reader cannot verify the central minimality claim without doing extra work. The secondary issue in Corollary 1 (soficity of the restriction to the F_infinity subgroup) is a standard preservation property and does not threaten correctness. No internal contradiction or circularity was found. I therefore agree with the CONDITIONAL verdict: the construction is plausible and likely correct, but the separation lemma needs to be made precise.","tokens_in":10518,"tokens_out":33649,"duration_ms":366545,"concrete_test":"Re-derive the separation argument in Lemma 16 as an induction on the number of equal coordinate pairs. For the induction step, fix a pair x_f = x_f' with d = f' f^{-1} != 1, pick b in B with [d,b] nontrivial (exists by the injectivity assumption); using antidiagonal minimality of A, choose a so that the common value y of the class of f lies in Supp([d,b]) and so that for every currently distinct pair (i,j), the images under the b-twists remain distinct (avoiding finitely many graphs {c_i u = c_j v}). Prove that these choices are compatible (each bad set is nowhere dense), that the chosen pair becomes distinct, and that distinctness of already separated pairs is preserved. If this induction can be written out, Theorem 5 stands; if not, Lemma 16 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 16 (Section 7), the minimality proof for the A*B-subaction on X^D reduces to: given a finite tuple x in X^F with equal coordinate pairs, find w in A*B making all coordinates distinct, then use antidiagonal minimality of A to move into any open set. The paper only shows how to separate a single equal pair: choose b in B so that [phi_{f' f^{-1}}, b] has nonempty support, then choose a in A putting the common value y = a x_f = a x_f' in that support. It then asserts 'It is clear that by choosing a suitably, we can ensure that no new coincidences are introduced by applying b' and stops. This is not immediate when there are several equality classes or classes of size > 2: a must simultaneously place each class's common value in the support of the relevant commutator (for the chosen b) and place values of distinct classes outside finitely many graphs {c_i u = c_j v} to prevent new coincidences. The text does not prove such an a exists, nor does it explain how to iterate the step to eliminate all equal pairs while preserving previously separated ones. Since Lemma 16 is the engine of Theorem 5 and hence Theorem 1, this missing argument is load-bearing. A separate, smaller issue is Corollary 1's assertion that the induced shift restricted to (F4 x F2) rtimes F_infinity is sofic; this is a standard preservation fact but is not stated. The main concern remains the incomplete separation claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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∞.","tokens_in":10726,"tokens_out":25962,"duration_ms":294755,"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":[{"comment":"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.","section":"Lemma 16 (Section 7)"}],"minor_comments":[{"comment":"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'.","section":"Lemma 16, proof"},{"comment":"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.","section":"Corollary 1"},{"comment":"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":"Theorem 5, item 3"},{"comment":"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.","section":"Section 1.2"}],"recommendation":"major_revision","confidential_remarks":"Major revision seems appropriate. The stress-test concern is real: the 'It is clear' passage in Lemma 16 is the only serious gap, and it is fixable. The paper is within the journal's scope and I found no circularity or novelty issue. I would ask the authors to add the full separation argument, including the choice of b avoiding finitely many centralizers, and to state the preservation facts used in Corollary 1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper gives the first example of a non-finitely-generated group admitting a minimal sofic shift, answering an open question of Doucha, Melleray and Tsankov. The group is explicit: (F4 x F2) ⋊ F∞. The construction is genuinely new, combining Thompson's V action's antidiagonal minimality with minimal Φ-joinings, then using induction to a semidirect product and simulation theory to get soficity. It is not a routine application of existing tools, and the paper is honest about what it leaves open (e.g., the SFT version).\n\nThe high-level architecture is sound. The reliance on earlier self-simulability results is appropriate, and those are independent published theorems. I see no circularity.\n\nThe soft spot is exactly what the stress-test note flags: Lemma 16. After showing you can separate a single equal pair, the proof says 'It is clear that by choosing a suitably, we can ensure that no new coincidences are introduced by applying b.' That is not clear. With multiple equality classes, or classes of size greater than two, you need to choose a to simultaneously place each class's common value in the right open set and avoid finitely many graphs of the form u = c_j v. Antidiagonal minimality gives a lot of freedom, so I suspect the claim is true, but it is a genuine combinatorial step and it is the engine of the whole theorem. This needs a real argument, not a single sentence.\n\nThe smaller issue is Corollary 1: the claim that the induced shift on the F∞-subgroup is sofic is standard, but it is not stated or justified. That is a one-line fix.\n\nBottom line: the result is significant and the proof is probably correct, but as written it is not fully checkable because Lemma 16 hides the load-bearing step. I would send it to a good symbolic dynamics journal and require the author to expand Lemma 16 and add the short justification in Corollary 1. After that, I would expect it to be accepted.","headline":"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.","tokens_in":11312,"tokens_out":2335,"would_cite":true,"duration_ms":25428,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["minimal sofic shift","non-finitely generated group","Thompson's V","antidiagonal minimality","semidirect product","symbolic dynamics","simulation theorem","free group"],"falsifier":"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.","tokens_in":10222,"feed_emoji":"🌀","tokens_out":9347,"duration_ms":92375,"temperature":0.7,"pith_summary":"The paper proves that a countable group which is not finitely generated can still admit an infinite minimal sofic shift, resolving in the affirmative a question posed by Doucha, Melleray and Tsankov. The group is explicitly built as $(F_4 \\times F_2) \\rtimes F_\\infty$, a semidirect product in which the free group factor acts by conjugation. The shift is obtained by inducing a sofic system on $F_4 \\times F_2$ to this semidirect product, and minimality of the induced $F_4$-action is forced by an antidiagonally minimal action of Thompson's group $V$. This matters because known obstructions rule out such shifts on non-finitely generated amenable or locally finite groups, leaving open whether any non-finitely generated group could work; the construction shows such groups exist.","feed_headline":"A minimal sofic shift exists on a non-finitely generated group","feed_subtitle":"The group is (F4 × F2) ⋊ F∞; Thompson's V supplies the transitivity that makes the shift minimal.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Poses the question answered by Theorem 1 and proves that amenable and locally finite non-finitely-generated groups cannot admit infinite minimal sofic shifts, supplying the target and the obstruction to beat.","marker":"[18]"},{"why":"Provides the self-simulability result (Lemma 1) that turns computable actions on products of nonamenable groups into sofic shifts; this is what makes the constructed shift sofic.","marker":"[8]"},{"why":"Proves Thompson's V is 3/2-generated, so a free group F2 can act on Cantor space through the natural antidiagonally minimal action of V.","marker":"[16]"},{"why":"Supplies the alternative simulation theorem for direct products used in Lemma 17 when the trivial-acting factor C is a direct product of infinite groups.","marker":"[5]"}],"fun_headline_variants":["Non-finitely generated group hosts minimal sofic shift","Minimal sofic shift found on non-finitely generated group","Thompson's V helps build minimal sofic shift on exotic group","Open question answered: minimal sofic shift on non-fg group","Group (F4×F2)⋊F∞ admits minimal sofic shift"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Non-finitely generated group hosts minimal sofic shift","Minimal sofic shift found on non-finitely generated group","Thompson's V helps build minimal sofic shift on exotic group","Open question answered: minimal sofic shift on non-fg group","Group (F4×F2)⋊F∞ admits minimal sofic shift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1417,"prompt_tokens":794,"completion_tokens":623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":533}},"tokens_in":410,"tokens_out":623,"duration_ms":5558,"temperature":1.0,"reasoning_tokens":533,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:02:54.824517+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dense and comeager conjugacy classes in zero-dimensional dynamics, 2025","cited_arxiv_id":null,"evidence_quote":"Poses the question answered by Theorem 1 and proves that amenable and locally finite non-finitely-generated groups cannot admit infinite minimal sofic shifts, supplying the target and the obstruction to beat."},{"cited_title":"Infinite 3 2 -generated groups","cited_arxiv_id":null,"evidence_quote":"Proves Thompson's V is 3/2-generated, so a free group F2 can act on Cantor space through the natural antidiagonally minimal action of V."},{"cited_title":"A geometric simulation theorem on direct products of finitely generated groups","cited_arxiv_id":null,"evidence_quote":"Supplies the alternative simulation theorem for direct products used in Lemma 17 when the trivial-acting factor C is a direct product of infinite groups."}],"review_version":1}