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On finite extensions of lamplighter groups

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One family of lamplighter extensions yields three never-before-seen combinations of algorithmic properties.

desk verdict Strong paper, three open questions answered; Theorems 1 and 2 are solid, but Theorem 4.1's appendix needs real proof before the third bullet is fully established. read the letter →

arxiv 2507.13203 v2 pith:4YW7CGWY submitted 2025-07-17 math.GR cs.DMcs.FL

Corentin Bodart This is my paper · ORCID
classification math.GRcs.DMcs.FL MSC 20F1020F6520E22
keywords lamplightergroupscentralextensionssubgroupmembershipproblemuniformwordconjugacygrowthseriesgeodesics
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

This paper studies a family of groups obtained from lamplighter groups $C_2 \wr H$ by adding one central element $z$ of order 2 and declaring that $[a,a^h]=z$ exactly when $h$ lies in a chosen subset $I \subset H$. The central claim is that this single family, by varying $H$ and $I$, realizes three combinations of algorithmic properties that were previously unknown: decidable membership in any fixed finitely generated subgroup with no uniform algorithm for membership across all such subgroups; rational volume growth series together with an undecidable word problem; and a decidable word problem with an undecidable conjugacy problem while the language of conjugacy geodesics (minimal-length representatives of conjugacy classes) is unambiguously context-free. If correct, these examples settle two open questions in the literature about how far such properties can be separated. The paper also characterizes residual finiteness and isomorphism within the family, and argues that the construction should be a standard tool for building groups with prescribed algorithmic behavior.

What carries the argument

The central object is the commutator-twisted lamplighter extension $G(H,I)$. Its nilpotent kernel is an explicit 2-step nilpotent group $N(H,I)\cong(\bigoplus_H \mathbb{F}_2)\times \mathbb{F}_2$ with product $(u,m)(v,n)=(u+v,\,m+n+\omega_I(u,v))$, where $\omega_I(u,v)=\sum_{g<h}\chi_I(g^{-1}h)u_g v_h \pmod 2$. This normal form turns each algorithmic question into a parity or membership question controlled by $I$: membership of $z$ in a subgroup becomes a divisibility condition, conjugacy of $x$ with $xz$ becomes a parity condition on $g^{-1}\mathrm{supp}(x)\cap I$, and the growth series is computed by the observation that, for the generating set $S=\{a,az,t^\pm,t^\pm z\}$, the projection to $C_2\wr\mathbb{Z}$ preserves word length outside $\{1,z\}$. The same split, an explicit nilpotent kernel sitting over a lamplighter quotient, lets the paper pull undecidability from $I$ into the group while inheriting the context-free geodesic language of $C_2\wr F_2$.

What would settle it

Enumerate all elements of $C_2\wr F_2$ of length at most 8 for a fixed free basis, compute the true minimal length in each conjugacy class, and check whether every word accepted by the grammar of Theorem A.4 (equivalently, every element satisfying conditions (1)--(4) of Proposition A.1) is one of those minimal representatives; a single accepted word that is not minimal would refute the context-free claim of Theorem 4.1.

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Extended reading notes

Core claim

Starting from a group $H$ and a symmetric subset $I\subset H$ with $1\notin I$, the paper defines $G(H,I)=\langle a,H,z \mid a^2=z^2=[a,z]=[h,z]=1,\ [a,a^h]=z \text{ if } h\in I,\ 1 \text{ otherwise}\rangle$, a central extension $1\to \langle z\rangle \to G(H,I) \to C_2\wr H \to 1$. The discovery is that this small twist of the lamplighter presentation is flexible enough to separate algorithmic problems that had resisted separation. Specifically, the paper constructs a recursive $I\subset \mathbb{Z}$ for which $G_I$ has decidable Subgroup Membership but undecidable Uniform Subgroup Membership; a non-recursive $I$ for which $G_I$ has undecidable word problem yet rational volume growth series with respect to a natural generating set; and a recursive $I\subset F_2$ for which $G(F_2,I)$ has decidable word problem, undecidable conjugacy problem, and an unambiguously context-free conjugacy-geodesic language. The subset $I$ acts as a switch: it controls where the extra commutator $z$ appears, and hence where undecidable instances hide, while the lamplighter quotient keeps the geometry and the geodesic language tractable.

Load-bearing premise

The load-bearing premise is that the quoted characterization of shortest representatives of conjugacy classes in $C_2\wr F_r$ is correct in the sufficiency direction, because the context-free grammar of Theorem A.4 is built on it and would accept non-geodesic words if that direction failed.

Editorial extensions

If this is right

  • The classical observation that recursively presented groups with undecidable word problem have non-computable growth series cannot be extended to all finitely generated groups, since $G_I$ has rational growth series and undecidable word problem.
  • Subgroup Membership and Uniform Subgroup Membership are genuinely different problems: within $G_I$, every fixed finitely generated subgroup has decidable membership, yet no algorithm can take a pair of generators and decide whether $z$ lies in the subgroup.
  • A conjugacy-geodesic language as low as context-free does not imply a decidable conjugacy problem; the group $G(F_2,I)$ has an unambiguously context-free conjugacy-geodesic language and undecidable conjugacy problem.
  • Within the family, residual finiteness is equivalent to $H$ being residually finite and $I$ being a union of cosets of a finite-index subgroup, and $G(H,I)\simeq G(H,J)$ exactly when some automorphism of $H$ carries $I$ to $J$, under the unit-conjecture hypothesis.

Reading between the lines

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

  • A natural next test, raised by the paper as Question 2.7, is whether a sufficiently irregular recursive $I$ makes Uniform Subgroup Membership decidable in $G_I$ while the Knapsack problem is undecidable; that would separate yet another pair of algorithmic problems inside the same family.
  • Because the rational growth of Theorem 2 comes from an isometry to a direct product, the generating-set trick may transfer to other central extensions with rational-growth quotients, producing more groups with rational growth series and undecidable word problem.
  • The context-free part of Theorem 4.1 rests on the sufficiency direction of a quoted characterization of conjugacy geodesics in $C_2\wr F_r$; a direct verification of that direction for short elements would remove the main residual doubt, and a mismatch would isolate exactly which part of the grammar construction fails.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies central extensions G(H,I) of lamplighter groups C2≀H, where the extension datum is a symmetric subset I⊂H. The main results are: (1) for a suitable recursive I⊂Z, the group GI has decidable Subgroup Membership but undecidable Uniform Subgroup Membership; (2) for a suitable recursive I⊂Z, GI can have undecidable Word Problem while its growth series with respect to a natural generating set is rational; and (3) for a suitable recursive I⊂F2, the group G(F2,I) has decidable Word Problem, undecidable Conjugacy Problem, and unambiguously context-free language of conjugacy geodesics. The paper also discusses residual finiteness, the co-Word Problem, and the Isomorphism Problem inside this class. Theorems 1 and 2 are supported by concrete Turing reductions and explicit computations, but the third theorem relies heavily on an appendix that characterizes conjugacy geodesics in C2≀Fr and asserts that ConjGeo(C2≀Fr,T) is unambiguously context-free.

Significance. If the main claims hold, the paper answers open questions of Duchin and Shapiro (growth series vs. undecidable Word Problem) and of Ciobanu, Hermiller, Holt and Rees (low-complexity ConjGeo with undecidable Conjugacy Problem), and it introduces a flexible family of central extensions of lamplighter groups as a toolkit for constructing groups with prescribed algorithmic properties. Theorems 1 and 2 are, in my reading, established: the reductions via Romanovskii's theorem, Lemma 2.3, and the isometry argument in Proposition 3.2 are convincing and largely self-contained. The paper is well written and carefully credits prior work, including Genevois's construction and Mercier's preprint. The main weakness is that the third theorem's ConjGeo claim is not proved at the required level of detail in Appendix A, and Lemma 4.5 as stated contains a language-theoretic error that needs correction. These are substantial but local issues, and in my view they are fixable within the manuscript's framework.

major comments (3)
  1. [§4.2, Lemma 4.5] The displayed equality ConjGeo(G,S) = τ^{-1}(ConjGeo(Q,T)) ∪ (F\{1}) is not correct as written. If τ is the erasing homomorphism S*→T* that sends every element of F to the empty word, then in the example G=G(Z,∅), Q=C2≀Z, T={a,t±}, S={a,az,t±,t±z,z}, the word w=za satisfies τ(w)=a∈ConjGeo(C2≀Z,T), so w belongs to the right-hand side. But the element \bar w=za is the generator az, which has S-length 1, while w has length 2, so w is not a conjugacy geodesic. The proof also states the equivalence 'w is a conjugacy geodesic iff ... τ(w) is a conjugacy geodesic' without the necessary length condition ℓ(w)=ℓ(τ(w)). The correct statement is that a word using no letters from F is a conjugacy geodesic exactly when its image under the non-erasing restriction S\F→T lies in ConjGeo(Q,T), together with the empty word and the length-one words for the non-identity elements of F; equivalently, one must add the condition ℓ(w)=ℓ(τ(w)). Since this lemma is the bridge from the quotient language to ConjGeo(G,S) in Theorem 4.1, the proof needs to be corrected accordingly.
  2. [§A.1, Proposition A.1] The sufficiency direction of Proposition A.1 is load-bearing for Theorem A.4 and is not proved. The sentence 'all the reductions made in [27,§3] to go from g satisfying (1-4) to a conjugacy geodesic actually preserve the length' is an appeal to an unpublished preprint and does not by itself establish that every element satisfying conditions (1)–(4) is length-minimal in its conjugacy class. Since the grammar in Theorem A.4 is supposed to generate exactly the language of conjugacy geodesics, this direction is essential. Please supply a complete proof, or state and prove the relevant result from Mercier's preprint in sufficient detail that the length-preservation claim can be checked.
  3. [§A.4, Theorem A.4] Theorem A.4 is the central technical support for the third bullet of Theorem 4.1, but its proof is only the assertion 'We claim ...'. First, the rule schemata such as Es ← s X1...Xℓ s^{-1} with 'the Xi are distinct elements' should be expanded into a genuine finite context-free grammar; this is in fact possible because the available variable set {a}∪{Ev | v∈B±} is finite, so the constraints on distinctness and on ℓ merely enumerate finitely many productions, but the text should say so explicitly. More importantly, there is no argument that the language generated by all the rules is exactly ConjGeo(C2≀Fr,T), and no argument for the claimed uniqueness of leftmost derivations. Given that the correctness of the grammar also depends on the unproved sufficiency direction of Proposition A.1, this is a substantial gap that must be closed before the third main theorem can be considered established.
minor comments (6)
  1. [§0.3] In the definition of conjugacy, 'there exists c∈G such that g = cgc^{-1}' should read 'g = chc^{-1}'.
  2. [§2, Lemma 2.3] The phrase 'halts after m steps' should be 'halts after exactly m steps', and m=0 should be excluded (or the definition arranged so that 0∉I), to avoid ambiguity about what 'after 0 steps' means.
  3. [§3, Proposition 3.4] The converse direction of Proposition 3.4 (if I is recursive then the growth series is computable) is only implicit; since I recursive gives decidable Word Problem by Theorem 1.4, one can enumerate all words up to length n and remove duplicates, but this should be stated explicitly.
  4. [§1, Theorem 1.4] The 'if' direction of Theorem 1.4 is described in words ('now we just have to move the factors around') and would benefit from a more formal normal-form argument; as written this is a decidability proof and the commutator bookkeeping is not fully spelled out.
  5. [§A, Theorem A.4] In the final paragraph of Theorem A.4, the claim refers to 'ConjGeo(C2 ≀ F2, T)' while the theorem statement is for C2≀Fr; this should be fixed to Fr.
  6. [§A, reference [27]] The paper relies essentially on Mercier's arXiv preprint [27]; since it is not peer-reviewed, the reliance should be stated explicitly, and if the preprint has since been published or revised, that information should be included.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main results are derived from explicit constructions and external benchmarks, not from their own conclusions.

full rationale

The derivation chain is self-contained against external benchmarks. The family G(H,I) is defined by a presentation involving I, but none of the theorems assumes the property it proves: Theorem 1 reduces Uniform Subgroup Membership undecidability to the halting problem via Lemma 2.3, and uses Romanovskii's theorem for the decidable half; Theorem 2 obtains rational growth by the explicit metric identity ||g||_S = ||tau(g)||_T for S = tau^{-1}(T) and combines it with Theorem 1.4's characterization of the Word Problem; Theorem 4.1's undecidable Conjugacy Problem is reduced to Lemma 4.3, while the ConjGeo statement is transferred from the quotient C2 wr F2 by Lemma 4.5 and the appendix's grammar analysis. The only load-bearing external imports (Mercier's characterization of conjugacy geodesics, Johnson's growth computation, Romanovskii's metabelian membership theorem, Sale's conjugacy search result) are independent of the present paper and are cited as such, not self-citations. Proposition A.1's sufficiency direction and the finite-grammar status of Theorem A.4 are asserted rather than fully formalized, and those are correctness or completeness risks rather than instances of circularity: the paper does not define ConjGeo in terms of the grammar, nor does it fit parameters to the claims it then 'predicts'. No fitted input is renamed as a prediction, and no load-bearing conclusion reduces to an author's own prior result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters or ad hoc entities. Its constructions are explicit presentations and halting-based subsets I and J. The central claims rely on standard or cited results in group theory and formal language theory; the only fragile cited input is Mercier's characterization of conjugacy geodesics, which is identified here as a domain assumption.

assumptions (6)
  • standard math Romanovskii's theorem: finitely generated metabelian groups have decidable Uniform Subgroup Membership.
    Used in Theorem 2.1(a) to decide membership in the projection π(H) inside C2 ≀ Z, which is metabelian.
  • standard math Johnson's formula for the rational growth series of C2 ≀ Z with standard generators.
    Used in Proposition 3.2 to conclude that the growth series of GI with S = τ^{-1}(T) is rational.
  • domain assumption Mercier's characterization of conjugacy geodesics in C2 ≀ Fr, as restated in Proposition A.1.
    Load-bearing for the grammar in Theorem A.4 and hence for the context-free claim in Theorem 4.1. The paper provides a sketch but refers to [27] for the 'only if' direction.
  • standard math Sale's solution of the Conjugacy Search Problem in C2 ≀ Z.
    Used in Theorem 4.7 to find a conjugator in the quotient when deciding the Conjugacy Problem for GI.
  • standard math Semi-linearity of Knapsack solution sets in co-context-free groups.
    Used in Proposition 5.3(b) to deduce that a context-free co-Word Problem forces I to be eventually periodic.
  • standard math C2 ≀ Z embeds in Thompson's group V, and embeddability in V implies context-free co-Word Problem.
    Used in Proposition 5.3(a) to prove that coWP(GI) is context-free when I is periodic.

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Pith. "Pith review of On finite extensions of lamplighter groups." pith.science (2026). https://pith.science/paper/4YW7CGWY

@misc{pith2026250713203,
  author       = {Pith},
  title        = {Pith review of: On finite extensions of lamplighter groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4YW7CGWY}},
  note         = {Machine review of arXiv:2507.13203}
}
read the original abstract

We study a family of groups consisting of the simplest extensions of lamplighter groups. We use these groups to answer multiple open questions in combinatorial group theory, providing groups that exhibit various combinations of properties: 1) Decidable Subgroup Membership and undecidable Uniform Subgroup Membership Problem, 2) Rational volume growth series and undecidable Word Problem and 3) Recursive (even context-free) language of conjugacy geodesics, decidable Word Problem, and undecidable Conjugacy Problem. We also consider the co-Word Problem, residual finiteness and the Isomorphism Problem within this class.

Figures

Figures reproduced from arXiv: 2507.13203 by the authors.

Figure 1
Figure 1. The third isomorphism theorem 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. An element g = (u, h) with h ̸= 1, and the different associated constructions. Vertices in supp(u) are marked in yellow. Proposition A.1. Consider the generating set T = {(δ1, 1)} ∪ B±, where B is a basis of Fr. An element g = (u, h) ∈ C2 ≀ Fr has minimal length in its conjugacy class if and only if it satisfies the following conditions: (1) supp(u) ⊆ wS0 ⊔ wc1S1 ⊔ wc1c2S2 ⊔ . . . ⊔ wc1 . . . ckSk, (2) T0 ∩ Tk ⊆ [1,… view at source ↗
Figure 3
Figure 3. A shorter element g ′ ∼ g, using the failure of (2). (3) If g doesn’t satisfy (3), we can cancel out some lamps in supp(u)∩[1, w] and supp(u) ∩ [h, hw] in the same ⟨h⟩ orbit, marked in red on [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: A shorter element g ′′ ∼ g ′ , using the failure of (3). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: A shorter element g ′′′ ∼ g ′′, using the failure of (4). For the “if” part, we observe that all the reductions made in [27, §3] to go from g satisfying (1-4) to a conjugacy geodesic actually preserve the length. Remark A.2. The statement can be simplified in rank r = …
Figure 6
Figure 6. Figure 6: Set where u can possibly be supported. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

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