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Conjugator Length in the Baumslag-Gersten Group

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Pith's one-line read The conjugator length function of the Baumslag-Gersten group lies between two towers of exponentials of logarithmic height, so it grows faster than any tower of exponentials of fixed height.

desk verdict New bounds on conjugator length in the Baumslag-Gersten and iterated Baumslag-Solitar groups; the results are likely correct, but the lower-bound proof has a repairable gap in the normal-form claim. read the letter →

arxiv 2507.21505 v1 pith:6THHQ53T submitted 2025-07-29 math.GR

Conan Gillis This is my paper · ORCID
classification math.GR MSC 20F1020F6520E06
keywords conjugatorlengthBaumslag-GerstengroupiteratedBaumslag-Solitarone-relatortowerofexponentialsannulardiagramspowercircuits
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 computes, up to a natural asymptotic equivalence, the conjugator length function of the Baumslag–Gersten group $G=\langle a,t \mid t a t^{-1} a (t a t^{-1})^{-1}=a^2\rangle$: it is at least $E(\lfloor (1/3)\log_2 n\rfloor - 1, 1)$ and at most $E(\lfloor \log_2 n\rfloor, 1)$, where $E(m,n)$ is the $m$-times iterated exponential. Because the height of the tower grows with $\log n$, $CL_G(n)$ grows faster than any tower of exponentials of fixed height. The paper also proves that the $m$-th iterated Baumslag–Solitar group $G_m$ has conjugator length $CL_{G_m}(n)\simeq E(m-1,n)$, which pins down the exact growth class in that case. Since conjugator length measures the length of a shortest element carrying one word to a conjugate word, this quantifies the difficulty of the conjugacy problem in $G$, paralleling the role of the Dehn function for the word problem.

What carries the argument

The argument is carried by annular diagrams for conjugate words, organized into corridors and rings labeled by the generators, together with a representation of integers as sums of the form $\sum_j n_j 2^{m_j}$; Corollary 4.3 converts the number of terms in such a sum into a lower bound on $|s_i^k|$. For the lower bound one uses the witness pair $u=s_1^2$ and $v=s_0^{E(m,n)-1}s_1^2$ and asserts that a minimal conjugator has the two-generator normal form $s_1^{\alpha}s_0^{\beta}$, so its length is controlled by $|s_0^{\beta}|$; the identity $3\sum_{j=0}^{E(m-1,n)/2}2^{-\alpha+2j}=2^{-\alpha}(E(m,n)-1)$ then feeds Corollary 4.3. For the upper bound, corridor and ring analysis together with the distortion estimates from [Pla04] for powers of $s_0$ and $s_1$ in $G$ bounds conjugators by $E(\lfloor \log_2 n\rfloor,1)$.

What would settle it

Exhibit, for some $m\ge 2$ and $n$, a reduced annular diagram witnessing the conjugacy of $s_1^2$ and $s_0^{E(m,n)-1}s_1^2$ in $G_m$ whose shortest connecting path uses an $s_2$-letter (or, in $G$, a $t$-letter) and contains no excisable ring; that would break the normal-form claim on which the $E(m-1,n)$ lower bound depends.

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Extended reading notes

Core claim

The central discovery is Theorem 1.4: for the Baumslag–Gersten group, $E(\lfloor (1/3)\log_2 n\rfloor - 1, 1) \preceq CL_G(n) \preceq E(\lfloor \log_2 n\rfloor, 1)$. The lower bound is produced from the witness pair $u_n=s_1^2$, $v_n=s_0^{E(m,n)-1}s_1^2$ inside a subgroup of $G$ isomorphic to $G_{2m}$, while the upper bound comes from reducing arbitrary conjugate words to cyclically Britton-reduced representatives and analyzing annular diagrams. Along the way Theorem 1.3 establishes the sharp statement $CL_{G_m}(n) \simeq E(m-1,n)$ for the $m$-th iterated Baumslag–Solitar group. The upshot is that the shortest conjugator in $G$ can be as large as an exponential tower whose height is logarithmic in the input length, and in the iterated-Baumslag–Solitar case the growth is an exact iterated exponential up to $\simeq$.

Load-bearing premise

The lower bound rests on the structural claim that a shortest conjugator between $s_1^2$ and $s_0^{E(m,n)-1}s_1^2$ in $G_m$ uses only the generators $s_1$ and $s_0$, so no $s_j$-letter with $j\ge 2$ (and no $t$-letter in $G$) can survive in a reduced annular diagram.

Editorial extensions

If this is right

  • For the $m$-th iterated Baumslag–Solitar group, conjugator length is exactly the $(m-1)$-fold iterated exponential up to $\simeq$, so each additional stable letter raises the growth class by one exponential level.
  • For the Baumslag–Gersten group, the worst-case conjugator length is at least a tower of height $\lfloor (1/3)\log_2 n\rfloor-1$ and at most a tower of height $\lfloor \log_2 n\rfloor$; in particular it exceeds every tower of exponentials of fixed height.
  • If the conjecture in Question 1.6 is correct, no one-relator group has a conjugator length function larger than that of $G$, and standard techniques would then give a solution to the conjugacy problem for all one-relator groups.
  • The upper bound passes through cyclically Britton-reduced representatives, so any conjugate pair in $G$ can be connected by a conjugator whose length is at most $E(\lfloor \log_2 n\rfloor,1)$ up to the equivalence.

Reading between the lines

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

  • The factor $1/3$ in the lower bound's height is probably an artifact of the proof: removing it would require only enlarging the input scale to $n^3$, and the same witness pairs may already realize the full $\lfloor \log_2 n\rfloor$ height.
  • Proving the asserted two-generator normal form in full generality would likely collapse the gap to $E(\lfloor \log_2 n\rfloor - C, 1) \preceq CL_G(n) \preceq E(\lfloor \log_2 n\rfloor, 1)$.
  • Theorem 1.3 suggests a general pattern for iterated HNN extensions with doubling relations: conjugator length is an iterated exponential whose height is the number of stable letters, and the Baumslag–Gersten group behaves like the infinite-iteration limit, which is why its height becomes logarithmic rather than constant.
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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

2 major / 6 minor

Summary. The paper studies the conjugator length function CL_G(n) of the Baumslag-Gersten group and CL_{G_m}(n) of the m-th iterated Baumslag-Solitar group. It proves the upper bound CL_G(n) ⪯ E(⌊log_2 n⌋, 1) and the lower bound E(⌊(1/3)log_2 n⌋ - 1, 1) ⪯ CL_G(n), together with the exact equivalence CL_{G_m}(n) ≃ E(m-1, n) for the iterated Baumslag-Solitar groups. The proofs combine annular-diagram and corridor techniques with signed-binary complexity estimates imported from power-circuit theory, and with external results on distortion and on conjugacy in the generic case.

Significance. If the results hold, they provide the first computation of the conjugator length function of the Baumslag-Gersten group up to the standard equivalence relation, showing that it grows faster than any fixed-height exponential tower, and they give an exact iterated-exponential bound for the conjugator length of iterated Baumslag-Solitar groups. The paper's architecture is coherent: the upper bounds are built from careful corridor arguments and established distortion results, and the lower-bound strategy via signed-binary complexity is natural and well matched to the group structure. The principal weakness is a missing Baumslag-Solitar normal-form argument in the lower-bound proofs; this is a load-bearing gap, but it is local and likely fixable.

major comments (2)
  1. [§4, Proposition 4.6] The proof asserts, without proof, that a minimal-length conjugator γ between u_n = s_1^2 and v_n = s_0^{E(m,n)-1}s_1^2 is equal in G_m to s_1^α s_0^β. The preceding reduction only shows that γ can be represented by a word on {s_0, s_1}; it does not establish the two-block normal form s_1^α s_0^β. In the subgroup ⟨s_0, s_1⟩ ≃ BS(1,2), the conjugators between s_1^2 and s_0^N s_1^2 are exactly the elements s_0^{-N/3} s_1^q for q ∈ Z, and the shortest among them is the one with q = 0. The proof needs a lemma showing that a shortest conjugator in G_m has this form, or at least that its length is bounded below by a constant fraction of |s_0^{-N/3}|_{G_m}. Without such a lemma, the application of Corollary 4.3 to the exponent β is unjustified, and the lower bound CL_{G_m}(n) ⪰ E(m-1,n) does not follow.
  2. [§6, Proposition 6.5 and Corollary 6.4] The lower bound for the Baumslag-Gersten group inherits the identical gap. Corollary 6.4 assumes that after excising non-contractible t-rings, the relevant conjugator has the form s_1^α s_0^β, and Proposition 6.5 invokes 'the same argument as Proposition 4.6' to finish the proof. The missing BS(1,2) normal-form argument must be supplied here as well, since the lower bound of Theorem 1.4 depends on the length of s_0^{ε-1} in G being at least the stated tower height.
minor comments (6)
  1. [§4, Proposition 4.6] The sentence 'By restricting attention to the subdiagram contained between Q and un' appears to be a typo: the subdiagram that yields a conjugator between u_n and v_n without s_2-edges is the one between Q and the boundary component labeled by v_n, not the one between Q and u_n.
  2. [§4, Proposition 4.6] There is a typo in 'non-contractibke s_2-ring'; it should be 'non-contractible s_2-ring'.
  3. [§3, Proposition 3.4] In the sentence 'since once of w_1, w_2 does not represent an element of ⟨s_{r-1}⟩_{G_m}', the word 'once' should be 'one'.
  4. [§6, Proposition 6.5] The expression 'w = s_1^2 - un' should read 'w = s_1^2 = u_n'; the current wording is confusing.
  5. [§5, Proposition 5.5] The statement 'Since ⟨s_0,s_1⟩_G is normal in G' is false: t s_1 t^{-1} = t^2 s_0 t^{-2} is not an element of ⟨s_0,s_1⟩_G. The subsequent case split does not actually need normality, but the sentence should be corrected or removed.
  6. [§5, Lemma 5.4] The quoted passage from [DMW16] is not typeset cleanly; the notation 'ta =_{BS(1,2)} a^2 t' should be replaced with a properly formatted display.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's upper and lower bounds are derived from internal diagrammatic arguments plus external benchmarks, and no prediction reduces to its own input by construction.

full rationale

I walked the derivation chain for Theorems 1.3 and 1.4 and found no step in which a claimed prediction or first-principles result reduces to its own inputs. The upper bounds for the iterated Baumslag-Solitar groups are built from internal corridor/pinch arguments in Proposition 3.4, with only the base case CL_BS(1,2)(n) ≃ n imported from the external reference [Sal16] and the linear conjugator length of Z taken as standard. The lower bounds are built from signed-binary complexity estimates (Corollary 4.3) whose central input is the external power-circuit lemma Lemma 2.8(2) of [MUW12], together with internally proved structural lemmas about word length and pinches; the witness elements s_1^2 and s_0^{E(m,n)-1}s_1^2 are shown to have combined length O(n) by Lemma 4.4, which is proved independently by induction from the defining relations rather than assumed from the conclusion. For the Baumslag-Gersten group, the upper bound relies on Platonov's distortion computation [Pla04], the external algorithm of [DMW16], and the linear CL of BS(1,2), all independent evidence; the lower bound inherits the same signed-binary machinery through Lemma 6.1, again without assuming the tower bound being proved. The paper contains no fitting of parameters to the target quantity and no load-bearing self-citation: [BRS25] is cited only as a survey and [Ril17] as exposition of Platonov's result. The skeptical observation that Proposition 4.6 asserts without proof that a minimal conjugator has normal form s_1^alpha s_0^beta is a potential correctness or completeness gap, but it is not circularity: that assertion is not an input to the definition of CL_{G_m}, and the argument does not define the conjugator length to be the claimed normal form. Since no equation-level reduction of the conclusion to an input was exhibited, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No fitted free parameters: this is pure mathematics with no data. The central claims rest on the paper's own corridor and pinch structural lemmas (2.16, 2.17, 2.24, 2.25), the asserted minimal-conjugator normal form (Proposition 4.6), and four imported external results ([MUW12] signed-binary term counts, [Pla04] distortion, [Sal16] CL of BS(1,2), [DMW16] conjugacy algorithm). Lemma 4.5's arithmetic error and Lemma 2.5's sketch mean two ledger entries have incomplete proofs as printed. No new entities are postulated, matching the pure-math content.

assumptions (9)
  • domain assumption The ⪯/≃ equivalence (Definition 1.1): multiplicative and additive constants plus linear argument inflation Cn+C are negligible; every theorem is stated up to this equivalence.
    Defines the yardstick of the paper. The relation is loose enough that constant-level errors (for example the Lemma 4.5 discrepancy between (E(m-1,n)-1)/4 and (E(m-1,n)-4)/4) are absorbed by the main theorems by construction.
  • domain assumption Lemma 2.16 corridor classification for reduced annular diagrams over G_m: top-rank s_k-edges occur only inside s_k-corridors, with no contractible rings, and the top rank is forced.
    Proved in Section 2.4, but it is the linchpin of the upper bound (Lemma 3.2, Proposition 3.4) and of the reduction in Proposition 4.6 that eliminates rank-2 letters from witness diagrams.
  • ad hoc to paper Every minimal conjugator between s_1^2 and s_0^{E(m,n)-1}s_1^2 has normal form s_1^α s_0^β; the analogous statement holds in G after t-ring excision (used via Corollary 6.4).
    Asserted in the proof of Proposition 4.6 ('A minimal length conjugator γ equals s_1^α s_0^β in G_m for some α, β') without the underlying BS(1,2) normal-form derivation; Proposition 6.5 inherits the same premise for G. Load-bearing for both lower bounds.
  • standard math [MUW12, Lemma 2.8(2)]: any power-of-2 sum with coefficients ±1 equal to Σ_{j=0}^p 2^{2j+α} has at least p terms.
    Imported without proof. This is the engine of Corollaries 4.3 and 6.2, which convert word length of s_0^k into the number of signed binary terms of k, and through it both tower-scale lower bounds. Not verified in this review beyond spot checks for small p.
  • standard math Platonov distortion: for s_0^n in G, |n| is bounded by C E(⌊log₂(C|s_0^n|_G + C)⌋, 1) + C (Lemma 5.1, Corollary 5.2).
    Imported from [Pla04] via [Ril17]; used to bound the exponents n_i of t-pinch contents in Lemma 5.3, which feeds the upper bound for G. The paper notes the analogous statement for s_1 follows with a constant shift.
  • standard math CL of BS(1,2) is linear: CL_BS(1,2)(n) ≃ n [Sal16].
    Used as the base case m=1 of Theorem 1.3 and as the linear bound for conjugators inside ⟨s_0,s_1⟩ in Proposition 3.4 Case 1, Proposition 5.5, and Lemma 3.3.
  • standard math The [DMW16, Theorem 3] generic-case algorithm conjugates cyclically Britton-reduced u,v outside ⟨s_0,s_1⟩ using s_0^k with |k| ≤ |u| + |v| (Lemma 5.4).
    Imported algorithm plus a short verification using the Z[1/2]⋊Z model. Load-bearing for the generic half of the upper bound for G (Proposition 5.5).
  • domain assumption Lemma 2.17: ⟨t^{-m}s_0t^m, ..., s_0, ..., t^m s_0 t^{-m}⟩_G is isomorphic to G_2m with a bijection on chosen generating sets; hence |x|_H_m equals a shifted word length in G_2m (Lemma 2.20).
    Proved in Section 2.5; I checked the negative-index relation s_{-1}s_{-2}s_{-1}^{-1} = s_{-2}^2 follows from the defining relation. This transfer is what lets lower bounds for G_m apply to G in Corollary 6.4 and Proposition 6.5.
  • domain assumption Lemma 2.5 excision: identifying two non-crossing same-labeled loops of equal winding number and deleting the interposed band never increases minimal annular diameter.
    Presented as a proof sketch deferred to Lyndon-Schupp pp. 150-151. Used in Proposition 3.4 Case 2 and Proposition 5.5 to remove repeated t-rings and s_k-rings; its diameter claim is not fully demonstrated.

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Pith. "Pith review of Conjugator Length in the Baumslag-Gersten Group." pith.science (2026). https://pith.science/paper/6THHQ53T

@misc{pith2026250721505,
  author       = {Pith},
  title        = {Pith review of: Conjugator Length in the Baumslag-Gersten Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6THHQ53T}},
  note         = {Machine review of arXiv:2507.21505}
}
abstract

We show that the conjugator length function of the Baumslag-Gersten group is bounded above and below by a tower of exponentials of logarithmic height -- in particular it grows faster than any tower of exponentials of fixed height. We conjecture that no one-relator group has a larger conjugator length function than the Baumslag-Gersten group. Along the way, we also show that the conjugator length function of the $m$-th iterated Baumslag-Solitar groups is equivalent to the $m$-times iterated exponential function.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Conjugator length in finitely presented groups

    math.GR 2026-07 accept novelty 6.0 of 10

    The conjugator length function is quadratic for the integral Heisenberg group and Stallings' group, and cyclic-subgroup distortion can be promoted to conjugator length.

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Works this paper leans on

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