{"id":"d27aede3-a8ac-4f1c-bcc8-e2afda6bb865","arxiv_id":"2502.03602","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"One-relator groups with at least three generators are never periodically rigid, and quasi-planar groups are rigid exactly when virtually cyclic or torsion-free virtually Z^2.","lead":"This paper proves that every one-relator group with at least three generators is not periodically rigid: each admits a symbolic system with weak but not strong aperiodicity, confirming a special case of Bitar's 2024 conjecture. It also characterizes period-rigidity for quasi-planar groups and introduces a new transfer tool for finite-index subgroups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2's proof treats the m=1, G1=Z^2 branch as automatically periodically rigid, but the theorem's rigid class is torsion-free virtually Z^2; torsion virtually Z^2 groups such as D_∞×Z are left unhandled and would break the iff statement if rigid.","rationale":"I read the central Theorem 1 as making a precise claim and checked the supporting lemmas: Lemma 11 correctly uses the homomorphism |·|_c to apply Barbieri's criterion; Lemma 13's rewriting terminates by decreasing the total positive exponent sum; Lemma 12 and the right-extension Lemma 9 are standard and internally consistent. The theorem's proof does not appear to have a soft spot. The one load-bearing gap I find is in Corollary 2, exactly where the Reader places it: the m=1, G1=Z2 branch forgets the torsion-free hypothesis. Since the quasi-planar theorem is an advertised consequence and its 'only if' direction relies on this branch, the gap is material. It is fixable by a citation or construction, so I keep the CONDITIONAL verdict unchanged.","tokens_in":13841,"tokens_out":20334,"duration_ms":182061,"concrete_test":"Analyze the boundary case G=D_∞×Z with the finite-index torsion-free subgroup H=Z^2. Using Bitar [7, Proposition 6.9] (flagged in Remark 1), produce an SFT on G that is weakly but not strongly aperiodic. If such an SFT exists, the proof's line 'G1=Z2, so G is periodically rigid' in Corollary 2 is false as stated and the corollary needs an explicit torsion-free hypothesis; if no such SFT exists, D_∞×Z is a counterexample to Bitar's Conjecture 4 and the corollary's statement itself fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Corollary 2 (Section 5), after applying MacManus's Theorem 14 the authors split into cases. The case 'm=1, G1=Z2' ends with 'G is periodically rigid'. This is exactly where the statement's torsion-free hypothesis is needed and not verified. A quasi-planar group can be virtually Z^2 and have torsion, e.g. D_∞×Z has finite-index subgroup Z×Z and contains the reflection in D_∞. If such a group were periodically rigid, the 'only if' direction of Corollary 2 (and Bitar's Conjecture 4) would fail; the expected non-rigidity therefore requires a construction or citation. The paper's Remark 1 points to [7, Proposition 6.9] for behavior involving a torsion-free subgroup inside a group with torsion, but Corollary 2 does not invoke it. As written, the proof silently assumes all virtually Z^2 groups are rigid in this branch. This gap does not affect Theorem 1, whose proof chain (Freiheitssatz, Piantadosi, Cohen, Barbieri, Lemma 13) appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies periodically rigid finitely generated groups, i.e., groups in which every weakly aperiodic subshift of finite type (SFT) is also strongly aperiodic. The main theorem states that every finitely generated group admitting a one-relator presentation with at least three generators and a cyclically reduced relator is not periodically rigid (Theorem 1). The proof combines the Freiheitssatz, a Magnus-Moldavansky rewriting argument (Lemma 13), free extensions of SFTs, and known results of Piantadosi, Cohen, and Barbieri. The paper also proves that periodic rigidity is inherited by finite-index subgroups via a new \"right extension\" construction (Lemma 9/Proposition 3), and uses this together with MacManus's structure theorem for quasi-planar groups to derive a classification of periodically rigid quasi-planar groups (Corollary 2): they are exactly the virtually cyclic or torsion-free virtually Z^2 groups.","tokens_in":14051,"tokens_out":32175,"duration_ms":257232,"significance":"If correct, Theorem 1 establishes Bitar's conjecture for a large and natural class of one-relator groups, and Corollary 2 resolves it for all quasi-planar groups. The right-extension lemma is a new and potentially useful tool for transferring non-rigidity from finite-index subgroups to overgroups, and its proof is detailed and appears sound. The paper relies on external theorems transparently, and no circularity is evident. The main issue is a gap in the proof of Corollary 2 concerning torsion in the virtually Z^2 case; this gap is local and fixable, but it must be addressed before the classification of quasi-planar groups can be considered established.","major_comments":[{"comment":"In the m=1, G1=Z^2 case, the proof concludes \"G is periodically rigid\" and stops. This is the assumption of the \"only if\" direction, not the required conclusion, which is \"torsion-free virtually Z^2.\" A quasi-planar group can be virtually Z^2 with torsion; for example, D_∞ × Z is virtually Z^2 and contains the reflection of D_∞. If such a group were periodically rigid, the statement of Corollary 2 would fail. The proof needs an additional argument showing that every virtually Z^2 group with torsion is not periodically rigid, for instance by citing [7, Proposition 6.9] (as in Remark 1) or Bitar's classification of virtually nilpotent groups. As written, the \"only if\" direction of Corollary 2 is incomplete.","section":"Section 5, Corollary 2"}],"minor_comments":[{"comment":"The case split \"either G1=Z^2 or G1 is a surface group of genus at least 2\" is not exhaustive if \"surface group\" in MacManus's theorem includes non-orientable surfaces or the genus-0 (trivial) group. These cases are covered by the theorem's conclusion (virtually cyclic or torsion-free virtually Z^2), but the proof should say so explicitly.","section":"Section 5, Corollary 2"},{"comment":"The termination of the Magnus-Moldavansky rewriting process is asserted rather than demonstrated; stating the decreasing invariant (e.g., the sum of the positive exponent sums of the relator) would make the proof easier to verify.","section":"Section 4, Lemma 13"},{"comment":"The appeal to [7, Proposition 6.9] is terse; a short statement of the proposition would help the reader understand when the free extension loses strong aperiodicity.","section":"Section 2, Remark 1"},{"comment":"The statement should explicitly require G to be finitely generated (or at least have finite generating set S), to align with the SFT framework used in the paper and with the abstract.","section":"Theorem 1"},{"comment":"There are several typographical issues: \"period-ridigity\" in Section 5, \"occurences\" in Section 4, and a few other small typos. The authors should proofread the final version.","section":"Throughout"},{"comment":"Reference [22] is a MathOverflow post; if a peer-reviewed or final version of the result exists, it would be preferable to cite that.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central construction (Lemma 9) and Theorem 1 appear sound and are significant contributions. The only substantive concern is the missing torsion argument in Corollary 2's proof; this is easily fixable by citing known results on virtually Z^2 groups with torsion. I expect the paper to be acceptable after a minor revision that closes this gap and tightens the exposition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Do you know this paper? Quick take: the main theorem is real, the quasi-planar corollary has a fixable gap.\n\nWhat's actually new: Theorem 1, that every one-relator group with at least three generators is not periodically rigid. The proof chains Magnus's Freiheitssatz with Piantadosi's theorem and Cohen's end obstruction, and the key trick is Lemma 13, a Magnus-Moldavansky rewriting that either gives infinitely many ends or a relator where some generator has total exponent zero. Then Lemma 11's homomorphism argument kicks in. I went through the chain and it holds up. The right-extension construction (Lemma 9) is a genuinely useful addition: it transfers a weakly-but-not-strongly aperiodic SFT from a finite-index subgroup up to the group, and the proof is explicit and correct. That's a tool people will reuse.\n\nThe soft spot is Corollary 2. In the m=1 case with G1=Z^2, the proof says 'G is periodically rigid' without checking that G is torsion-free. A quasi-planar group can be virtually Z^2 and still have torsion, e.g., D_∞ × Z. The theorem's rigid class is torsion-free virtually Z^2; the proof silently assumes all virtually Z^2 groups are rigid. That's exactly the gap the stress-test note flags, and it's real. It doesn't affect Theorem 1, but it means the corollary as written overclaims. The fix should be straightforward: cite Bitar's Proposition 6.9 (or the relevant construction) to handle the torsion case, or add a sentence saying those groups are non-rigid. Without that, the paper advertises more than it proves.\n\nOverall: the citation pattern is clean, no circularity, the external results are genuinely external. The paper is a serious contribution to the Bitar-conjecture program. I'd send it to a referee, with a note about the quasi-planar section. After a minor-to-moderate revision it should be in good shape.","headline":"Solid special case of Bitar's conjecture for one-relator groups with a new finite-index transfer lemma, but the quasi-planar corollary has a torsion gap that needs patching.","tokens_in":14626,"tokens_out":7891,"would_cite":true,"duration_ms":66318,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","37B10","20F05","37B50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every finitely generated one-relator group with at least three generators is not periodically rigid: it carries a weakly aperiodic subshift of finite type that is not strongly aperiodic.","keywords":["one-relator groups","subshifts of finite type","periodic rigidity","weak aperiodicity","strong aperiodicity","quasi-planar groups","free products","finite-index subgroups"],"falsifier":"Test the mechanism on a concrete one-relator group: take $G=\\langle a,b,c \\mid a^2 b^3 c^5 \\rangle$, where $c$ appears with total exponent $5 \\neq 0$, so the proof rewrites the presentation and then applies the free-extension argument from the free subgroup generated by $a$ and $b$. Build the free-extension SFT of a weakly aperiodic SFT on $\\langle a,b\\rangle$ and check directly whether it is weakly but not strongly aperiodic on $G$, as predicted. For the quasi-planar corollary, an equally concrete test is the group $D_\\infty \\times \\mathbb{Z}$ (the direct product of the infinite dihedral group with $\\mathbb{Z}$): if it is periodically rigid, the stated 'torsion-free virtually $\\mathbb{Z}^2$' dichotomy would be false.","tokens_in":13598,"feed_emoji":"🧩","tokens_out":15546,"duration_ms":129419,"temperature":0.7,"pith_summary":"The paper proves that every finitely generated group with a one-relator presentation using at least three generators is not periodically rigid: it admits a nonempty subshift of finite type in which every configuration has an infinite orbit, yet some configuration is invariant under a nontrivial group element. This provides a large family for a recently proposed characterization, namely that the periodically rigid finitely generated groups are exactly the virtually cyclic groups and the torsion-free virtually $\\mathbb{Z}^2$ groups. The proof works by lifting a weakly aperiodic subshift from a free subgroup of rank two and then showing that the lift cannot be made strongly aperiodic, using a homomorphism that counts a generator's exponent in the relator. A second transfer result, for finite-index subgroups, turns the one-relator theorem into a full classification for quasi-planar groups, whose Cayley graphs are quasi-isometric to planar graphs.","feed_headline":"One-relator groups with 3+ generators are never periodically rigid","feed_subtitle":"Every such group carries a weakly aperiodic SFT that is not strongly aperiodic, confirming a 2024 conjecture.","key_machinery":"Two transfer constructions and one numerical invariant carry the proof. The free extension lifts a subshift of a subgroup $H$ to a subshift on $G$ by forcing every left $H$-coset to obey the same forbidden patterns; it preserves weak aperiodicity. The right extension, defined when $H$ has finite index, labels right cosets and preserves both weak and strong aperiodicity, which yields the finite-index heredity result. The numerical invariant is the homomorphism $|\\cdot|_c \\colon G \\to \\mathbb{Z}$ counting the total exponent of a generator $c$ in a word. When $c$ has exponent zero in the relator, the classical Freiheitssatz makes the subgroup generated by the other generators free of rank at least two; no conjugate of a nontrivial power of $c$ lies in that free subgroup, and this non-normality is exactly what prevents the free extension from being strongly aperiodic. The rewriting process re-expresses the group so that this situation holds, or shows that the group has infinitely many ends.","core_discovery":"The paper's central result is that a group $G$ admitting a presentation $\\langle S \\mid r\\rangle$ with $|S| \\ge 3$ and $r$ cyclically reduced and nonempty is not periodically rigid. Concretely, $G$ carries a nonempty subshift of finite type in which every configuration has infinite orbit, yet some configuration is fixed by a nontrivial element of $G$. The dichotomy in the proof is: either some generator has total exponent zero in $r$, in which case the remaining generators span a free subgroup of rank at least two and the free extension of its weakly aperiodic SFT is weakly but not strongly aperiodic on $G$; or no generator has exponent zero, in which case a rewriting procedure either creates such a generator while preserving the one-relator presentation, or reveals that $G$ has infinitely many ends, which alone prevents strong aperiodicity. Combining this with the finite-index heredity result and the structure theorem for quasi-planar groups yields the full classification: a quasi-planar group is periodically rigid exactly when it is virtually cyclic or torsion-free virtually $\\mathbb{Z}^2$.","pith_inferences":["The exponent-counting homomorphism is portable: any group with a free subgroup of rank at least two and a homomorphism that is nonzero on that subgroup's complement could be shown non-rigid by the same free-extension argument.","The finite-index heredity result points to a direct way to settle the unresolved torsion case in the quasi-planar proof: build a weakly-but-not-strongly aperiodic SFT on a virtually-$\\mathbb{Z}^2$ group with torsion, such as $D_\\infty \\times \\mathbb{Z}$.","The bottleneck for two-generator one-relator groups is not the free-extension step but the rewriting step, which reduces the generator count; extending the conjecture to all one-relator groups likely needs a new way to force the exponent-zero condition without losing a generator.","Because the proof never uses hyperbolicity or planarity, the same dichotomy may hold for any class where a structure theorem expresses groups as free products of rigid and non-rigid factors."],"forward_implications":["Every one-relator group with at least three generators admits a weakly but not strongly aperiodic SFT, so none of these groups is periodically rigid.","Period rigidity is inherited by finite-index subgroups: whenever a group is periodically rigid, every finite-index subgroup is periodically rigid too.","The periodic-rigidity conjecture holds for quasi-planar groups, including surface groups and groups whose Cayley graphs are quasi-isometric to planar graphs.","Virtually free-by-cyclic groups satisfy the conjecture: the low free-rank cases are the known rigid groups, and higher free rank yields a weakly-but-not-strongly aperiodic SFT."],"supporting_citations":[{"why":"supplies the Freiheitssatz: the subgroup generated by all but one generator is free when that generator occurs in the relator.","marker":"[33]"},{"why":"shows a free group of rank at least two admits a weakly aperiodic SFT, the seed configuration for the construction.","marker":"[36]"},{"why":"proves the free extension of a weakly aperiodic SFT from a subgroup to a supergroup remains weakly aperiodic.","marker":"[28]"},{"why":"gives the criterion under which such a free extension fails to be strongly aperiodic, applied via the exponent homomorphism.","marker":"[4]"},{"why":"establishes that groups with infinitely many ends admit no strongly aperiodic SFT, handling that branch of the dichotomy.","marker":"[11]"},{"why":"classifies quasi-planar groups as virtually free products of free and surface groups, converting Theorem 1 into the quasi-planar result.","marker":"[32]"},{"why":"formulates the periodic-rigidity conjecture and supplies the known rigidity of virtually cyclic and torsion-free virtually $\\mathbb{Z}^2$ groups.","marker":"[7]"},{"why":"ensures a free product with at least two nontrivial factors contains a free group of rank two, used for the many-factor case.","marker":"[37]"}],"fun_headline_variants":["One-relator groups with 3+ generators lack period rigidity","3-generator one-relator groups never periodically rigid","Non-rigid: one-relator groups with ≥3 generators","Periodic rigidity fails for one-relator groups with 3+ gens","No period rigidity in one-relator groups with 3 or more generators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quasi-planar proof's case where one factor is $\\mathbb{Z}^2$ declares the group periodically rigid without first checking torsion-freeness, so the classification rests on the unstated premise that a virtually-$\\mathbb{Z}^2$ quasi-planar group with torsion is never periodically rigid.","fun_headline_variants_meta":{"raw":{"variants":["One-relator groups with 3+ generators lack period rigidity","3-generator one-relator groups never periodically rigid","Non-rigid: one-relator groups with ≥3 generators","Periodic rigidity fails for one-relator groups with 3+ gens","No period rigidity in one-relator groups with 3 or more generators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000131,"raw_usage":{"total_tokens":1113,"prompt_tokens":913,"completion_tokens":200,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":111}},"tokens_in":529,"tokens_out":200,"duration_ms":2033,"temperature":1.0,"reasoning_tokens":111,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:23:57.724545+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the mechanism on a concrete one-relator group: take $G=\\langle a,b,c \\mid a^2 b^3 c^5 \\rangle$, where $c$ appears with total exponent $5 \\neq 0$, so the proof rewrites the presentation and then applies the free-extension argument from the free subgroup generated by $a$ and $b$. Build the free-extension SFT of a weakly aperiodic SFT on $\\langle a,b\\rangle$ and check directly whether it is weakly but not strongly aperiodic on $G$, as predicted. For the quasi-planar corollary, an equally concrete test is the group $D_\\infty \\times \\mathbb{Z}$ (the direct product of the infinite dihedral group with $\\mathbb{Z}$): if it is periodically rigid, the stated 'torsion-free virtually $\\mathbb{Z}^2$' dichotomy would be false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the Freiheitssatz: the subgroup generated by all but one generator is free when that generator occurs in the relator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"shows a free group of rank at least two admits a weakly aperiodic SFT, the seed configuration for the construction."},{"cited_title":"Barbieri","cited_arxiv_id":null,"evidence_quote":"gives the criterion under which such a free extension fails to be strongly aperiodic, applied via the exponent homomorphism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that groups with infinitely many ends admit no strongly aperiodic SFT, handling that branch of the dichotomy."},{"cited_title":"Realizability of Subgroups by Subshifts of Finite Type","cited_arxiv_id":"2406.04132","evidence_quote":"formulates the periodic-rigidity conjecture and supplies the known rigidity of virtually cyclic and torsion-free virtually $\\mathbb{Z}^2$ groups."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"ensures a free product with at least two nontrivial factors contains a free group of rank two, used for the many-factor case."}],"review_version":1}