Some Orbits of Free Words that are Determined by Measures on Finite Groups

Chen Meiri; Doron Puder; Liam Hanany

Some Orbits of Free Words that are Determined by Measures on Finite Groups

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every positive integer d, the words x^d and [x,y]^d are determined, up to automorphism of the free group, by the measures they induce on all finite groups.

desk verdict Solid, careful paper that proves new profinite rigidity results for x^d and [x,y]^d; the main theorems hold up, with Khelif's theorem as the one heavy external input. read the letter →

arxiv 1908.03801 v2 pith:JRNDQBG6 submitted 2019-08-10 math.GR

Liam Hanany , Chen Meiri , Doron Puder This is my paper · ORCID

classification math.GR MSC 20E0520E1820E3620P05

keywords freegroupswordmapsfiniteprofiniterigidityAut(F)orbitsmeasuresexpectedfixedpointscommutatorwords

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

Every word in a free group induces a probability measure on every finite group by plugging in independent uniform random elements. The paper addresses the open question whether two words that induce the same measure on every finite group must lie in the same orbit of the automorphism group of the free group. It proves a new special case: for every positive integer d, the power x^d and the commutator power [x,y]^d are universally profinitely rigid, meaning any word with the same finite-group measures as one of them is automorphic to it. This matters because it turns a probabilistic condition on all finite groups into a clean combinatorial conclusion about the shape of the word.

What carries the argument

The main tool is the expected number of fixed points Tr_w(N) of a w-random permutation in the symmetric group S_N, where the word w is evaluated on independent uniformly random permutations. The paper generalizes the known fixed-point comparison: for a word w not contained in a proper free factor and free words u_1,...,u_k that do not generate a free factor, Tr_{w(u_1,...,u_k)}(N) is strictly larger than Tr_w(N) for all large N. This inequality is obtained by decomposing the count of fixed points through an algebraic-extension partial order on subgroups and an inversion over that order. It is the step that converts equality of measures into statements about freeness and free factors, and ultimately into automorphism equivalence.

What would settle it

A concrete falsifier would be a word w that induces exactly the same measure as x^d or [x,y]^d on every finite group but is not an Aut(F)-image of it. For x^d, one could look for a non-primitive u such that u^d matches x^d on all symmetric groups; the paper predicts the expected fixed-point count of u^d is strictly larger than that of x^d for large N, so a candidate showing equality for all large N would disprove the theorem.

Watch

Extended reading notes

Core claim

The central result is that the measure profile of x^d or [x,y]^d on finite groups is rigid enough to determine the word up to automorphism. The power case is handled by showing that profinite rigidity is inherited by powers: if w is profinitely rigid, then w^d is too, using the fact that the set of dth powers is closed in the profinite topology and that roots of a power in the profinite completion already lie in the free group. The commutator case uses the theorem that a word whose image in every finite quotient is a commutator is itself a commutator, and then applies a fixed-point asymptotic comparison to force the two entries of the commutator to generate a free factor. That free factor is therefore automorphic to the standard basis element pair x and y.

Load-bearing premise

The commutator half of the proof depends on the cited theorem that a word whose image is a commutator in every finite quotient is itself a commutator; if that theorem failed, the reduction of the [x,y] case would not go through.

Editorial extensions

If this is right

For every d≥1, the words x^d and [x,y]^d are universally profinitely rigid.
If a word induces the same measure as x^d on every finite group, it is an Aut(F)-image of x^d; the same holds for [x,y]^d.
The Aut(F)-orbit of each of these words is closed in the profinite topology of the free group.
If a word lies in the same Aut(F-hat)-orbit as x^d or [x,y]^d, then it already lies in the same Aut(F)-orbit, so no new words arise from profinite automorphisms.
Profinite rigidity is preserved under powers: any power of a profinitely rigid word is profinitely rigid.

Reading between the lines

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

The same fixed-point comparison may identify other words with a unique minimal algebraic extension as profinitely rigid, offering a natural next class beyond x^d and [x,y]^d.
Measures on symmetric groups alone cannot decide the orbit of plain commutator words: the paper notes that [x,y] and xyx^{-1}y^{-1} induce the same S_N-measure for every N while lying in different Aut(F)-orbits, so the full force of all finite groups is essential.
The profinite-centralizer argument for powers suggests a route to prove rigidity of higher powers of any word whose centralizer in the free group is cyclic.
One could test the method on surface words or on words like x^d y^d, where the algebraic-extension rank data are tractable, to see whether measure equality forces automorphic equivalence there as well.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

I read this through and largely agree with the reader's take. The paper extends the known primitive case of profinite rigidity to all powers of a generator and all powers of the commutator. Theorem 1.4 is genuinely new and, as far as I can tell, correct. The two technical ingredients are also worth having: Theorem 1.7 shows profinite rigidity survives taking powers, via profinite centralizers and Herfort–Ribes, and Theorem 1.5 is a clean quantitative strengthening of the Puder–Parzanchevski fixed-point inequality. I checked the reduction of Theorem 1.4 to Theorems 1.5 and 1.7 and found no circularity or missing case.

The main soft spot is the external black box. The commutator case leans on Khelif's theorem that a word which is a commutator in every finite quotient is itself a commutator. That is a published, standard-sounding result, and the stress-test note confirms it is applied correctly, but it does mean the paper's central new claim is not self-contained. The reliance is real and load-bearing, yet I don't see it as a flaw, only a boundary of the method. A second, minor point is that Theorem 1.7 as stated is about rigidity in a fixed free group; the universal version follows by applying it in free extensions, which is immediate but could have been spelled out. The paper also honestly notes that [x,y] and xyxy^{-1} have the same measures on symmetric groups while lying in different orbits, so the symmetric-group measure alone cannot distinguish the commutator orbit. That caveat is good to see in print.

Where does this leave the value? It is not ground-shaking: Conjecture 1.2 remains open and this only extends the known list of rigid orbits, but it is a substantive, clean step. The proofs are detailed and appear complete. The paper is written for people working on word maps, profinite rigidity, or free groups, and they will want it on record. I would send it to a serious referee without hesitation. My own verdict is close to the reader's: accept, with only the mild suggestion that the authors make the role of the external theorems more prominent in the introduction. It deserves to be in the literature.

For peer review: yes, this paper should be refereed, and it will likely pass with minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper studies the following instance of Conjecture 1.2: if two words in a free group induce the same probability measure on every finite group, must they lie in the same Aut(F)-orbit? The authors prove (Theorem 1.4) that for every d≥1, the words x^d and [x,y]^d are universally profinitely rigid: any word with the same measures on all finite groups is Aut(F)-equivalent to x^d (respectively to [x,y]^d). The proof has two main ingredients: Theorem 1.5, a trace inequality for the expected number of fixed points of w-random permutations that generalizes the main result of Puder–Parzanchevski [PP15], and Theorem 1.7, which states that profinite rigidity is preserved under taking powers. The commutator case is reduced by Khelif's theorem (Theorem 1.9) to the trace inequality, and the power case is reduced via Lubotzky's theorem on the profinite closure of the set of dth powers. The paper also includes a self-contained proof (Proposition 4.5) that measures on symmetric groups alone suffice for the primitive-power case.

Significance. If correct, the result provides the first infinite families beyond primitive words for which the measure-equivalence problem of Conjecture 1.2 has a positive answer. The trace inequality of Theorem 1.5 is a clean and potentially reusable quantitative generalization of [PP15]'s main theorem; the paper notes in Remark 3.7 that it applies to surface words as well. The reduction arguments are clearly presented, and the main external dependencies (Khelif's Theorem 1.9, Lubotzky's Theorem 4.1, Nica's cycle-count asymptotics, Herfort–Ribes centralizer theorem) are published and correctly invoked. The paper is careful to distinguish the main proof from the auxiliary proof in §4.2. The proofs are detailed and, apart from a few local presentation issues, appear complete.

minor comments (5)

[§4.2, proof of Theorem 4.8] The step 'Taking expectations and then taking the limit as N→∞' requires more than the convergence in distribution supplied by Nica's theorem, because c_t(φ(w)) is not uniformly bounded. The authors should justify, for example by invoking uniform integrability or convergence of factorial moments, that the limits of the first two moments exist and depend only on the power b. This point does not affect the proof of the main theorem, since Theorem 4.8 is only used for the auxiliary Proposition 4.5.
[§4.2, Lemma 4.7] The formula E[c_{bt}(σ)^2] = 1/(bt) + 1/(b^2 t^2) for N≥2bt is used without proof or reference. It is a standard moment fact for cycle counts in a uniform random permutation, but a one-line justification or a citation would improve self-containment.
[Remark 2.6] The implication diagram is difficult to read in the typeset version; in particular, the notation for the relation 'w1 \overline{AutF}∼ w2' appears without its distinguishing bar. Please clarify the notation so that the two occurrences of 'AutF∼' in the diagram are visually distinct.
[Throughout] There are numerous broken exponents and spacing artifacts in the text, such as 'wd' in Theorem 1.7, 'F act 1.1', and 'natur al' in the abstract. These appear to be typesetting issues and should be corrected in the final version.
[Corollary 1.10] The phrase 'its image w′∈Q is in the support of the w′-measure' is correct because the quotient image is a value of the word map, but the sentence could be expanded slightly so that the reader does not confuse the support of the measure with the set of elements that are values of the specific word w′ as a group element.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is based on independent published theorems and newly proved inequalities, not on its own conclusions.

full rationale

The derivation chain is non-circular. The main theorem (Theorem 1.4) is reduced in Section 1 to two cases: primitive powers, which are covered by the previously published primitive-word theorem of [PP15] (and [Pud14]), and the commutator word [x,y], which is handled via Khelif's external theorem (Theorem 1.9). Khelif's theorem is used to show that a word inducing the same measures as [x,y] is itself a commutator; this is an independent published result whose statement does not include the target conclusion. The subsequent step, applying Theorem 1.5 to force the two commutator factors to form a free basis, is proved in Sections 3.1-3.2 from algebraic-extension theory and the Möbius-inversion machinery of [PP15], and it is a genuinely new inequality rather than a restatement of the conclusion. Theorem 1.7 is explicitly proved using only profinite-completion and free-product facts (Herfort-Ribes and Lubotzky/Thompson's closed-powers theorem), not word measures, so it is not a circular prediction. No fitted parameters are introduced, no quantity is renamed as a prediction, and the self-citations to [PP15] supply prior independent results that do not themselves assert the rigidity of x^d or [x,y]^d. The external dependencies, including Khelif's theorem, are published, parameter-free results with stated assumptions that do not assume the paper's main claims. Thus no circular step meets the required evidentiary standard, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new entities and no fitted numbers. The central assertion is a theorem derived from standard group theory and cited specializations of prior results. The main axioms are external theorems (Khelif, Lubotzky, Herfort-Ribes, Nica, PP15); if any of these were false, the proof would break, but they are all peer-reviewed results. No ad hoc assumption is introduced for this paper.

assumptions (8)

domain assumption Khelif's theorem: if the image of w in every finite quotient of F is a commutator, then w is a commutator.
Invoked in Section 1 (Theorem 1.9) to turn measure equality with [x,y] into the statement w=[u,v]; external result from [Khe04].
domain assumption Lubotzky's theorem: the set {u^d | u in F} is closed in the profinite topology on F.
Used in the proof of Theorem 1.7 (Section 4.1) to conclude that a word with the same measures as w^d is itself a dth power; cited from [Tho97] and re-proven in Theorem 4.8 via Nica's theorem.
domain assumption Herfort-Ribes Theorem B: for a free profinite product A (square cup) B, the centralizer of a in A is contained in A.
Used in Lemma 4.3 to show the centralizer of a word in the profinite completion is the closure of its centralizer in F; external result [HR85].
domain assumption Nica's theorem: the limiting distribution of cycle counts of a free word in random permutations depends only on the root (the non-power u in w=u^b).
Used in the proof of Theorem 4.8 (Section 4.2) to establish Lubotzky's theorem; cited from [Nic94].
domain assumption PP15 results: the Mobius-inverted function R_{H,J}(N) is rational and has asymptotic N^{1-rank J} + O(N^{-rank J}) (Theorem 3.4), and the primitive word theorem (Theorem 1.8).
Foundation for the proof of Theorem 3.6 and therefore Theorem 1.5; the present paper cites [PP15] rather than reproving these statements.
domain assumption Proposition 3.1 (MVW07): algebraic extensions of subgroups of free groups satisfy the stated properties, including unique factorization into an algebraic extension followed by a free factor.
Used throughout Section 3, especially Eq. (7) and the expansion in Eq. (10).
domain assumption Hall's theorem: a non-power u in F can be extended to a basis of a finite-index subgroup H <= F.
Used in Lemma 4.3 to decompose the closure of H as a free profinite product.
standard math Standard properties of profinite completions and free profinite products (RZ10), such as the isomorphism that the free profinite product of completions is the completion of the free product.
Used in Lemma 4.3 and Theorem 2.2; see [RZ10].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Some Orbits of Free Words that are Determined by Measures on Finite Groups." pith.science (2026). https://pith.science/paper/JRNDQBG6

@misc{pith2026190803801,
  author       = {Pith},
  title        = {Pith review of: Some Orbits of Free Words that are Determined by Measures on Finite Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRNDQBG6}},
  note         = {Machine review of arXiv:1908.03801}
}

abstract

Every word in a free group $F$ induces a probability measure on every finite group in a natural manner. It is an open problem whether two words that induce the same measure on every finite group, necessarily belong to the same orbit of $\mathrm{Aut}F$. A special case of this problem, when one of the words is the primitive word $x$, was settled positively by the third author and Parzanchevski [arXiv:1202.3269]. Here we extend this result to the case where one of the words is $x^d$ or $\left[x,y\right]^{d}$ for an arbitrary $d\in\mathbb{Z}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

[1]

Characters and solutions to equations in finite groups

Alon Amit and Uzi Vishne. Characters and solutions to equations in finite groups. J. Algebra Appl. , 10(4):675--686, 2011

work page 2011
[2]

Automorphism-invariant positive definite functions on free groups

Beno\^ i t Collins, Michael Magee, and Doron Puder. Automorphism-invariant positive definite functions on free groups. In Proceedings of the 27th International Conference in Operator Theory (OT27) , 2020. to appear, available at arXiv:1906.01518

work page Pith review arXiv 2020
[3]

Word measures on symmetric groups

Liam Hanany and Doron Puder. Word measures on symmetric groups. In preparation, 2020

work page 2020
[4]

Torsion elements and centralizers in free products of profinite groups

Wolfgang Herfort and Luis Ribes. Torsion elements and centralizers in free products of profinite groups. J. Reine Angew. Math. , 358:155--161, 1985

work page 1985
[5]

Finite approximation and commutators in free groups

Anatole Khelif. Finite approximation and commutators in free groups. Journal of Algebra , 281(2):407--412, 2004

work page 2004
[6]

Word maps and spectra of random graph lifts

Nati Linial and Doron Puder. Word maps and spectra of random graph lifts. Random Structures and Algorithms , 37(1):100--135, 2010

work page 2010
[7]

R. C. Lyndon and P. E. Schupp. Combinatorial group theory . Springer-Verlag, 1977

work page 1977
[8]

Word measures on unitary groups

Michael Magee and Doron Puder. Word measures on unitary groups. preprint arXiv:1509.07374 v2, 2016

arXiv 2016

Show all 22 references

[9]

Matrix group integrals, surfaces, and mapping class groups I : U (n)

Michael Magee and Doron Puder. Matrix group integrals, surfaces, and mapping class groups I : U (n) . Inventiones Mathematicae , 218(2):341--411, 2019

2019
[10]

Matrix group integrals, surfaces, and mapping class groups II : O (n) and S p(n)

Michael Magee and Doron Puder. Matrix group integrals, surfaces, and mapping class groups II : O (n) and S p(n) . preprint arXiv:1904.13106, 2019

1904 arXiv
[11]

Surface words are determined by word measures on groups

Michael Magee and Doron Puder. Surface words are determined by word measures on groups. Israel Journal of Mathematics , 2020. to appear, available at arXiv:1902.04873

2020 arXiv
[12]

J. A. Mingo, P. \'S niady, and R. Speicher. Second order freeness and fluctuations of random matrices. II . U nitary random matrices. Adv. Math. , 209(1):212--240, 2007

2007
[13]

Algebraic extensions in free groups

Alexei Miasnikov, Enric Ventura, and Pascal Weil. Algebraic extensions in free groups. In Geometric group theory , pages 225--253. Springer, 2007

2007
[14]

On the number of cycles of given length of a free word in several random permutations

Alexandru Nica. On the number of cycles of given length of a free word in several random permutations. Random Structures & Algorithms , 5(5):703--730, 1994

1994
[15]

Measure preserving words are primitive

Doron Puder and Ori Parzanchevski. Measure preserving words are primitive. Journal of the American Mathematical Society , 28(1):63--97, 2015

2015
[16]

Primitive words, free factors and measure preservation

Doron Puder. Primitive words, free factors and measure preservation. Israel J. Math. , 201(1):25--73, 2014

2014
[17]

Asymptotics of word measures on GL _n( F _q)

Doron Puder and Danielle West. Asymptotics of word measures on GL _n( F _q) . In preparation, 2020

2020
[18]

Profinite groups , volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete

Luis Ribes and Pavel Zalesskii. Profinite groups , volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete . Springer, second edition, 2010

2010
[19]

Some results and problems in the theory of word maps

Aner Shalev. Some results and problems in the theory of word maps. In L. Lov \'a sz, I. Ruzsa, V.T. S \'o s, and D. Palvolgyi, editors, Erd\" o s Centennial (Bolyai Society Mathematical Studies) , pages 611--650. Springer, 2013

2013
[20]

Thompson

John G. Thompson. Power maps and completions of free groups and of the modular group. Journal of Algebra , 191(1):252--264, 1997

1997
[21]

Profinite groups , volume 19 of London Mathematical Society, Monographs, New Series

John Stuart Wilson. Profinite groups , volume 19 of London Mathematical Society, Monographs, New Series . Oxford University Press, 1998

1998
[22]

Essential surfaces in graph pairs

Henry Wilton. Essential surfaces in graph pairs. Journal of the American Mathematical Society , 31(4):893--919, 2018

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Characters and solutions to equations in finite groups

Alon Amit and Uzi Vishne. Characters and solutions to equations in finite groups. J. Algebra Appl. , 10(4):675--686, 2011

work page 2011

[2] [2]

Automorphism-invariant positive definite functions on free groups

Beno\^ i t Collins, Michael Magee, and Doron Puder. Automorphism-invariant positive definite functions on free groups. In Proceedings of the 27th International Conference in Operator Theory (OT27) , 2020. to appear, available at arXiv:1906.01518

work page Pith review arXiv 2020

[3] [3]

Word measures on symmetric groups

Liam Hanany and Doron Puder. Word measures on symmetric groups. In preparation, 2020

work page 2020

[4] [4]

Torsion elements and centralizers in free products of profinite groups

Wolfgang Herfort and Luis Ribes. Torsion elements and centralizers in free products of profinite groups. J. Reine Angew. Math. , 358:155--161, 1985

work page 1985

[5] [5]

Finite approximation and commutators in free groups

Anatole Khelif. Finite approximation and commutators in free groups. Journal of Algebra , 281(2):407--412, 2004

work page 2004

[6] [6]

Word maps and spectra of random graph lifts

Nati Linial and Doron Puder. Word maps and spectra of random graph lifts. Random Structures and Algorithms , 37(1):100--135, 2010

work page 2010

[7] [7]

R. C. Lyndon and P. E. Schupp. Combinatorial group theory . Springer-Verlag, 1977

work page 1977

[8] [8]

Word measures on unitary groups

Michael Magee and Doron Puder. Word measures on unitary groups. preprint arXiv:1509.07374 v2, 2016

arXiv 2016