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Conjugator length in finitely presented groups

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Pith's one-line read This paper proves that the conjugator length functions of the 3-dimensional integral Heisenberg group and of Stallings' group grow quadratically, and it promotes the systematic study of this invariant.

desk verdict A genuinely useful survey, but the upper-bound proof for Stallings' group (Thm 4.14) has a real gap: the set of all conjugators in H is misdescribed when a free-group component is a proper power. read the letter →

arxiv 2607.20401 v1 pith:PJKVKTLG submitted 2026-07-22 math.GR

Martin R. Bridson , Timothy R. Riley , Andrew W. Sale This is my paper
classification math.GR MSC 20F6520F1020F06
keywords conjugacyproblemconjugatorlengthfunctionHeisenberggroupStallings'annulardiagramsHNNextensionsDiophantineequationsgeometrictheory
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 aims to make the conjugator length function — the shortest word needed to conjugate one element to another when they are conjugate — a standard quantitative invariant for the conjugacy problem, on a par with the Dehn function for the word problem. It proves two new benchmark results: in the 3-dimensional integral Heisenberg group and in Stallings' group, the conjugator length function grows quadratically. The Heisenberg proof reduces the problem to a single linear Diophantine equation in the two parameters of a candidate conjugator and applies a classical bound on the size of its smallest solution; the Stallings proof uses annular diagrams and a trimming lemma for HNN extensions to reduce to the same type of arithmetic estimate. For a sympathetic reader, the payoff is a concrete toolkit — annular diagrams, stable-letter corridors, Diophantine bounds — plus explicit examples that calibrate the difficulty of the conjugacy problem.

What carries the argument

Annular diagrams are the central objects: finite planar 2-complexes shaped like a cylinder whose boundary cycles read the two conjugate words u and v, with a path across the annulus carrying a conjugator w; Lemma 3.4 shows CL(u,v) is the least length of such a crossing. In HNN extensions, stable-letter t-corridors and t-annuli decompose these diagrams, and Corollary 3.7(iii) — the excision lemma — allows the authors to replace u,v by words of no greater length whose diagram has only radial t-corridors and no essential t-annuli. The second load-bearing tool is Lemma 4.10, a sharp bound on minimal solutions of a linear Diophantine equation Ax+By=C, which turns the search for short conjugators

What would settle it

Produce, for every constant C, a conjugate pair u,v in Stallings' group or the Heisenberg group with |u|+|v| ≤ n and shortest conjugator length > C n^2. In the Heisenberg case the search is concrete: write elements in normal form a^α b^β c^γ and solve the single linear equation (6); the theorem predicts a solution with |x|,|x̂| bounded by the coefficients, so a counterexample pair violating that bound falsifies the upper bound. For Stallings, the corresponding check is finding a pair that cannot be trimmed by the excision lemma without lengthening the words.

Watch

Extended reading notes

Core claim

The paper's central new results are Theorems 4.9 and 4.14: the conjugator length function of the 3-dimensional integral Heisenberg group H_3(Z) and of Stallings' group both grow quadratically. For a finitely generated group, CL(n) is the least upper bound on the length of a shortest word w with uw = wv in G, taken over all conjugate words u,v with |u|+|v| ≤ n. In the Heisenberg group, conjugacy of two elements in normal form forces the two off-diagonal parameters to match and leaves a single linear Diophantine equation in the two unknown parameters of a conjugator; an elementary bound on the smallest solution of such an equation gives the O(n^2) upper bound. The matching lower bound is witne

Load-bearing premise

The quadratic upper bound for Stallings' group rests on the excision lemma (Corollary 3.7(iii)): every conjugate pair of words can be replaced, without increasing lengths, by a pair whose annular diagram has only radial stable-letter corridors and no essential annuli; if that trimming step fails, the reduction to the free-product estimate and hence the O(n^2) bound collapses.

Editorial extensions

If this is right

  • In the Heisenberg group, the conjugacy search problem admits conjugators of length O(n^2); by Remark 4.11 the same holds for every higher-dimensional integral Heisenberg group.
  • Stallings' group now has a complete quadratic conjugator length function up to the standard equivalence, adding a classic exotic group to the short list of groups for which this invariant is known exactly.
  • The amalgamation construction of Theorem 5.2 shows that any distortion function of an infinite cyclic subgroup yields a lower bound on conjugator length, so the quadratic floor combines with distorted subgroups to produce groups with conjugator length at least as large as any prescribed distortion.
  • The survey's spectrum — linear for free, hyperbolic, BS(1,m), and mapping class groups, quadratic for Heisenberg and Stallings, non-recursive in other cases — makes conjugator length a discriminating quantitative invariant for the conjugacy problem.

Reading between the lines

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

  • The Heisenberg lower-bound witness suggests a general route for class-2 nilpotent groups: compute CL as the minimal norm of a solution to the defining system of linear Diophantine equations; the paper's survey already shows polynomial degrees of all integers are attainable, so one could test whether every class-2 nilpotent group has CL bounded by such a solution norm.
  • Because conjugator length is not a quasi-isometry invariant, the quadratic benchmarks imply that no coarse-geometric invariant alone can predict conjugator length; the paper's examples show index-two subgroups can differ in solvability of the conjugacy problem, so one can expect equally stark quantitative differences in finitely presented groups with identical large-scale geometry.
  • Theorem 2.1's equivalence with the width of free homotopies in a Riemannian manifold means numerical experiments on the Heisenberg nilmanifold could estimate CL by measuring minimal basepoint sweep; a measured quadratic width would independently corroborate the quadratic benchmark.
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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 / 4 minor

Summary. This paper is a survey of conjugator length functions for finitely presented groups, with several original results: quadratic growth for the integral Heisenberg group (Theorem 4.9) and for Stallings' group (Theorem 4.14), an elementary linear bound for BS(1,m) (Theorem 4.13), and constructions promoting cyclic subgroup distortion to conjugator length (Theorems 5.2 and 5.3). The paper develops the annular-diagram toolkit in Section 3, including a reduction lemma for HNN-extensions (Corollary 3.7), and surveys the known landscape. The Heisenberg proof is clean and fully displayed; the Stallings upper bound reduces to a Diophantine estimate for a claimed description of all conjugators in F(a,b) x F(c,d), and that description is not correct as written.

Significance. If the Stallings proof is repaired, the paper supplies two explicit quadratic benchmarks for conjugator length, the quantitative invariant attached to the conjugacy problem, complementing the theory of Dehn functions. The survey is well organized, and the diagrammatic foundations (annular diagrams, t-corridors, excision arguments) are useful for researchers entering the area. The proofs are self-contained modulo standard lemmas (van Kampen's lemma, annular diagram characterizations, Bezout-type Diophantine bounds), and the authors are explicit about the limitations of the hypothesis in Theorem 5.3.

major comments (2)
  1. [§4.5, proof of Theorem 4.14 (upper bound, displayed set W)] The assertion that W={θ1 θ2^p φ1 φ2^q | p,q∈Z} is the set of all w∈H with uw=wv in H is false in general. In the F(a,b) factor, take u_ab=a^2, v_ab=b^{-1}a^2b, and θ1=b. Then all conjugators from u_ab to v_ab are {a^k b : k∈Z}, but θ1 θ2^p = b(b^{-1}a^2b)^p = a^{2p}b, so conjugators such as a^{-1}b are omitted. Such an omitted conjugator can lie in K (e.g., it has z-length zero), so the word w_l obtained from the radial-corridor reduction need not lie in W. Consequently the Diophantine equation p z(θ2)+q z(φ2) = -z(θ1)-z(φ1) need not have a solution, and the O(n^2) upper-bound proof fails at this step. The proof should parametrize the actual set of conjugators using maximal roots in the free factors, then re-run the bounded-solution argument.
  2. [§2.4, definition of Width_M] Width_M(ℓ) is defined as an infimum over all pairs of loops of total length at most ℓ. With this definition Width_M is identically zero: take ρ0=ρ1 to be a constant loop. Thus Theorem 2.1 cannot hold as stated. The proof outline uses a worst-case width, and the intended definition should be a supremum over such pairs. Please correct the definition and adjust the surrounding text.
minor comments (4)
  1. [§3.3] Typo: 'jeodardises' should be 'jeopardizes'.
  2. [§4.2] Typo: 'seimhyperbolic' should be 'semihyperbolic'.
  3. [§4.5 / §5.3] In Corollary 3.7, 'not-corridor' should be 't-corridor'; near the end of the proof of Theorem 5.3, 's-corridor' should be 'r-corridor'.
  4. [§4.5] The notation S=H ˙∗_K is nonstandard; please state explicitly that this is the HNN-extension with stable letter t commuting with K, matching presentation (10). Also, the sentence justifying CL_H(n)≃n via CL_{A×B}≃max{CL_A,CL_B} could use a one-line explanation.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found; the central quadratic-conjugator-length proofs are carried out in the paper. The W-set concern in Theorem 4.14 is a potential proof gap, not circularity.

full rationale

I found no step in which a claimed prediction or first-principles result is equivalent, by construction or by definition, to its own input. The central new results, Theorem 4.9 and Theorem 4.14, are argued from explicit diagrammatic and Diophantine estimates: Theorem 4.9 derives the upper bound from a solvable linear Diophantine equation and proves the lower bound by an explicit family of conjugate words; Theorem 4.14 reduces the Stallings upper bound to the HNN set-up of Corollary 3.7, which is proved in the paper, and then to bounds on integer solutions of a linear equation. Corollary 3.7 itself is proved using van Kampen and annular diagram arguments, not assumed from prior work. Many survey statements cite the authors' prior papers, but those citations are expository or concern background results, and none of the load-bearing steps for the paper's novel theorems reduces to a self-citation. The skeptical counterexample about the set W in Section 4.5, if correct, would be a genuine mathematical error in the description of the full set of conjugators in H, and would invalidate that portion of the upper-bound proof; it is a correctness risk rather than a circularity, since the theorem is not being used as an input to its own proof. I therefore do not classify it as a circular step, and the circularity score remains low.

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

The paper introduces no fitted parameters and no new postulated physical or algebraic entities. The ζ-maps in §5.7 are defined mathematical tools, not independent entities. The central claims rest on standard combinatorial-group-theory theorems and a classical Diophantine bound, all identified above.

assumptions (6)
  • standard math Van Kampen's Lemma and annular diagram characterization (Lemma 3.2): conjugate words admit annular diagrams with boundary labels u,v, and CL(u,v) is the shortest cross-cut path in such a diagram (Lemma 3.4).
    Used throughout Sections 3–5 to translate conjugator length into diagram geometry; cited to Schupp and Lyndon–Schupp and proved in §3.1.
  • standard math Bound on solutions of binary linear Diophantine equations (Lemma 4.10): if Ax+By=C has an integer solution, it has one with |x|,|y| ≤ max{|A|,|B|,|C|}.
    Load-bearing for the quadratic upper bounds in Theorems 4.9, 4.14 and Remark 4.11; cited to [BFRT89, BR26a, Kor90].
  • standard math Normal form theorem for amalgamated free products and free products with amalgamation.
    Used in Theorem 5.2 to reduce arbitrary conjugators to Heisenberg words; invoked implicitly in the induction on L(w).
  • domain assumption t-corridor/t-annulus decomposition of annular diagrams over HNN-extensions, including excision of inessential annuli (Corollary 3.7).
    The Stallings group proof in §4.5 assumes all essential t-annuli can be excised and radial t-corridors pair t-letters; Corollary 3.7 is proved in §3.3 but rests on diagram-cutting/sewing operations.
  • domain assumption Finite generating sets and word-length conventions; CL is well-defined up to the ⪯ equivalence (Sections 1.1, 1.4, 2.2).
    Survey framework; all results are stated for finitely generated/presented groups with finite generating sets.
  • domain assumption Theorem 5.3 hypothesis CL_Λ ⪯ Dist^Λ_{⟨λ⟩}.
    The construction in §5.3 requires this hypothesis; authors note it is hard to verify in examples, limiting the theorem's utility.

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Pith. "Pith review of Conjugator length in finitely presented groups." pith.science (2026). https://pith.science/paper/PJKVKTLG

@misc{pith2026260720401,
  author       = {Pith},
  title        = {Pith review of: Conjugator length in finitely presented groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PJKVKTLG}},
  note         = {Machine review of arXiv:2607.20401}
}
abstract

The conjugator length function of a finitely generated group $G$ gives the minimal upper bound on the length of a conjugator for a pair of words that represent conjugate elements in $G$, as a function of the sum of the lengths of the words. Here, we seek to promote the systematic study of conjugator length functions by explaining their significance, by surveying what is known about them and by explaining fundamental techniques and examples.

Figures

Figures reproduced from arXiv: 2607.20401 by the authors.

Figure 1
Figure 1. Folding up a lollipop diagram to make an annular diagram to ⋆u by an edge-path labelled by w, w1, . . . , wm, respectively. We seek to form a van Kampen diagram ∆ for u over ⟨A | R ∪ {v}⟩ by folding up this lollipop diagram in the manner of a proof of van Kampen’s lemma as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Folding together a pair of edges But the proviso about planarity is a serious one: if the initial vertex of the edge e1 and the terminal vertex of e2 are the same, then identifying e1 and e2 will break the diagram’s planarity. The remedy is that in this circumstance, we instead remove the subdiagram that e1 and e2 together enclose, as shown on the right in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Take all the edges in this figure that are decorated with arrows to be t-edges. It then depicts an annular diagram with two radial t-corridors, one of length 0 and one of length 3, and one annular t-corridor, whose core loop is inessential. No t-corridor or t-annulus can cross itself or another t-corridor or t-annulus. Thus t-corridors and t-annuli partition diagrams into a collection of subdiagrams. Many arguments … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: , collapsing each disc to a vertex and each rectangle to an edge retracts Γˆ →→ Γ. We say that ρ is non-crossing when it lifts to a simple loop ˆρ in Γ. ˆ ρ ρˆ Γ = ∆(1) Γˆ ∆ (1) ρ [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: An example illustrating sewing a van Kampen diagram ∆ to fill the hole in an annular diagram Ω. In step I the diagram ∆ is inserted into the ‘hole’ and attached at one vertex. In step II two pairs of edges are folded together, and then two more pairs in step III. In st…
Figure 6
Figure 6. Figure 6: An illustration of t-corridors arising from a process of eliminating pinches in a word w. Now, for u and v of the lemma, define u ′ and v ′ to be words obtained in this manner. Then u ′ ∼ u ∼ v ∼ v ′ and conditions (a) and (b) are met, where for (b) we are appealing to…
Figure 7
Figure 7. Figure 7: Geodesic hexagons as arising in the proof of Lemma 4.5 in the case where q is on the side of H labelled by u1. Suppose q is on the side of H labelled by u1, which is the situation depicted in the figure. Express u1 as the concatenation u ′ 1u ′′ 2 of subwords, with the…
Figure 8
Figure 8. Figure 8: Geodesic hexagons as arising in the proof of Lemma 4.5 in the case where p and q are on the two sides of H labelled by w. Let w ′ = wiσ −1wˆj . Tracing paths through [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: An annular diagram in a hyperbolic group of defining relators, each instance of the latter replacing a non-geodesic subword σ of length at most (8δ + 1) by a geodesic word that equals σ in G. So the pairs u, ue and v, ve admit annular diagrams (see [PITH_FULL_IMAGE:fi…
Figure 10
Figure 10. Figure 10: The synchronous bi-combing condition This property is an essential feature of non-positive curvature in groups. It is enjoyed (for some k ≥ 0) by hyperbolic groups: wg can be taken to be any ge￾odesic representative of g, and the fellow-traveler property (f.t.p.) foll…
Figure 11
Figure 11. Figure 11: Conjugacy diagram arising from a bicombing Proof. Suppose u and v are words of total length n that represent conjugate ele￾ments of G. Then ug = gv in G for some g ∈ G. So uwg = wgv in G per the perimeter of the diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p022…
Figure 12
Figure 12. Figure 12: A van Kampen diagram for uw = wv in G. involving t, we see w1, . . . , wl ∈ K. We also have uwl = wlv in G and uwl = wlv in H = F(a, b) × F(c, d). Let z : H → Z be the map sending a, b, c, d to a fixed generator of Z. So K = Ker z. Let θ1 ∈ F(a, b), ϕ1 ∈ F(c, d) be su…
Figure 13
Figure 13. Figure 13: An annular van Kampen diagram over the group Σ. Suppose now that u and v together contain at least one letter p or q. Let Ω be an annular diagram of minimal area (that is, minimal number of faces) for u and v. Now, Σ is a multiple HNN-extension of Λ′ with stable lette…
Figure 14
Figure 14. Figure 14: Prohibited s-corridors in Ω. Case 3: Ω has a radial q-corridor but no p-corridor. Let ˆr = λr. Then Σ = ⟨ A ∪ λ, p, q, rˆ | R ∪  [λ, rˆ] , [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]

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