Period-rigidity of one-relator groups

Etienne Moutot; Sol\`ene J. Esnay; Ugo Giocanti

REVIEW 1 major objections 6 minor 38 references

Period-rigidity of one-relator groups

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.03602 v1 pith:62KUBW5N submitted 2025-02-05 math.GR cs.DMmath.DS

Solène J. Esnay , Ugo Giocanti , Etienne Moutot This is my paper · ORCID

classification math.GRcs.DMmath.DS MSC 20F6537B1020F0537B50

keywords one-relatorgroupssubshiftsoffinitetypeperiodicrigidityweakaperiodicitystrongquasi-planarfreeproductsfinite-indexsubgroups

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\mathbb{Z}^2$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Do you know this paper? Quick take: the main theorem is real, the quasi-planar corollary has a fixable gap.

What'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.

The 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.

Overall: 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.

Referee Report

1 major / 6 minor

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.

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 (1)

[Section 5, Corollary 2] 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.

minor comments (6)

[Section 5, Corollary 2] 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 4, Lemma 13] 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 2, Remark 1] 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.
[Theorem 1] 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.
[Throughout] 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.
[References] Reference [22] is a MathOverflow post; if a peer-reviewed or final version of the result exists, it would be preferable to cite that.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is derived from independent external results, and self-citations are contextual.

full rationale

The paper's derivation chain for Theorem 1 is self-contained relative to external theorems: Lemma 11 applies Magnus's Freiheitssatz (Theorem 10), Piantadosi's non-rigidity of free groups (Theorem 4), Jeandel's free-extension preservation of weak aperiodicity (Proposition 7), and Barbieri's criterion for non-strong aperiodicity (Theorem 8); none of these inputs is the target theorem or a fitted version of it. Lemma 13 is proved in-paper by Tietze transformations and a finite Magnus-Moldavansky rewriting process, and its conclusion (either infinitely many ends or a relator with zero exponent) is not an assumed prediction. Lemma 9's right-extension construction is explicit and verified directly, and it is used only to transfer non-rigidity from a finite-index subgroup to the ambient group. Corollary 2 combines Theorem 1 with MacManus's external classification of quasi-planar groups; the first direction is cited to Bitar's external work. The authors' own prior work [19] appears only as background for Baumslag-Solitar groups and is not load-bearing for the new results. The skeptical concern about the m=1, G1=Z^2 branch in Corollary 2 is a possible correctness gap regarding torsion in virtually Z^2 groups, but it is not a circularity: the branch does not reduce by definition to the target conclusion, nor is it justified by a self-citation chain.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters; nothing is fitted to data. The central proofs rely on external theorems: Magnus's Freiheitssatz, Piantadosi's non-rigidity of free groups, Cohen's no-strongly-aperiodic-SFT theorem for infinite ends, Barbieri's free-extension criterion, and MacManus's quasi-planar structure theorem. These are independent prior results, not outputs of this paper. The right-extension construction is new but proven in the paper.

assumptions (7)

standard math Freiheitssatz (Magnus, Theorem 10): For G=<S|r> with r cyclically reduced and s in S appearing in r, the subgroup generated by S\{s} is free of rank |S|-1.
Used in Lemma 11 to produce a free rank-2 subgroup <a,b> when a generator c appears in r with exponent sum 0.
standard math Theorem 4 (Piantadosi): Any free group of rank at least 2 is not periodically rigid.
Provides the weakly aperiodic SFT X on the free subgroup that is lifted in Lemmas 11 and 12.
standard math Theorem 6 (Cohen, patched by Salo and Genevois): A group with infinitely many ends cannot have a strongly aperiodic SFT.
Used in Lemma 12 to conclude the free extension cannot be strongly aperiodic.
standard math Theorem 8 (Barbieri): If there exists g in G\{1} such that no conjugate of a positive power of g lies in H\{1}, then the free extension of any SFT on H is not strongly aperiodic on G.
Used in Lemma 11 to show the lifted SFT is not strongly aperiodic via the homomorphism |.|_c.
standard math Theorem 14 (MacManus, Corollary D): A finitely generated group is quasi-planar iff it is virtually a free product of finitely many free groups and surface groups.
Basis of Corollary 2's classification.
standard math Arzhantseva-Minasyan-Osin result: A finitely generated group with infinitely many ends contains a free subgroup of rank 2.
Used in Lemma 12.
domain assumption Implicit fact: a one-relator group with at least 3 generators is neither virtually cyclic nor torsion-free virtually Z^2.
Not proven in the paper; used to reconcile Theorem 1's 'not periodically rigid' with the abstract's 'if and only if'.

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Cite this review

Pith. "Pith review of Period-rigidity of one-relator groups." pith.science (2026). https://pith.science/paper/62KUBW5N

@misc{pith2026250203602,
  author       = {Pith},
  title        = {Pith review of: Period-rigidity of one-relator groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/62KUBW5N}},
  note         = {Machine review of arXiv:2502.03602}
}

abstract

We follow in this paper a recent line of work, consisting in characterizing the periodically rigid finitely generated groups, i.e., the groups for which every subshift of finite type which is weakly aperiodic is also strongly aperiodic. In particular, we show that every finitely generated group admitting a presentation with one reduced relator and at least $3$ generators is periodically rigid if and only if it is either virtually cyclic or torsion-free virtually $\mathbb Z^2$. This proves a special case of a recent conjecture of Bitar (2024). We moreover prove that period rigidity is preserved under taking subgroups of finite indices. Using a recent theorem of MacManus (2023), we derive from our results that Bitar's conjecture holds in groups whose Cayley graphs are quasi-isometric to planar graphs.

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Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Arzhantseva, A

G. Arzhantseva, A. Minasyan, and D. Osin. The SQ-univers ality and residual properties of relatively hyperbolic groups. Journal of Algebra, 315(1):165–177, 2007

work page 2007

[2] [2]

Aubrun and M

N. Aubrun and M. Sablik. Simulation of eﬀective subshift s by two-dimensional subshifts of ﬁnite type. Acta Applicandae Mathematicae , 126(1):35–63, 2013

work page 2013

[3] [3]

Ballier and M

A. Ballier and M. Stein. The domino problem on groups of po lynomial growth. Groups, Geometry, and Dynamics , 12, 2013

work page 2013

[4] [4]

Barbieri

S. Barbieri. Aperiodic subshifts of ﬁnite type on groups which are not ﬁnitely generated. Proceedings of the American Mathematical Society , 151(09):3839–3843, 2023

work page 2023

[5] [5]

Bartholdi and V

L. Bartholdi and V. Salo. Shifts on the lamplighter group . arXiv e-print 2402.14508, 2024

arXiv 2024

[6] [6]

R. L. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 1966

work page 1966

[7] [7]

N. Bitar. Realizability of Subgroups by Subshifts of Fin ite Type. arXiv e-print 2406.04132, 2024

work page Pith review arXiv 2024

[8] [8]

N. Bitar. Subshifts of Finite Type on Groups : Emptiness and Aperiodic ity. PhD thesis, Universit´ e Paris-Saclay, 2024

work page 2024