Quasi-affine and quasi-quadratic maps of groups with non-abelian targets

Primoz Moravec

REVIEW 3 major objections 4 minor 16 references

Quasi-affine and quasi-quadratic maps of groups with non-abelian targets

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper proves that middle quasi-homomorphisms between groups with non-abelian targets are exactly constant perturbations of quasi-homomorphisms, and establishes a rigidity theorem for quasi-quadratic maps into torsion-free hyperbolic…

desk verdict Theorem 1.1 is a clean, real result, but the proof of Theorem 1.4 has a load-bearing gap at (4.9.4) that the authors need to fix. read the letter →

arxiv 2506.01577 v1 pith:3BEZ7U6G submitted 2025-06-02 math.GR

Primoz Moravec This is my paper

classification math.GR MSC 20F6520F67

keywords middlequasi-homomorphismsquasi-polynomialmapsquasi-quadraticmultiplicativequadruplestorsion-freehyperbolicgroupsrigidityconstructibility

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 "middle quasi-homomorphisms" — maps whose middle defect set $\{\varphi(x)^{-1}\varphi(xy)\varphi(y)^{-1}\}$ is bounded — are precisely the constant perturbations of ordinary quasi-homomorphisms. Equivalently, after right-multiplying by the inverse of the value at the identity, every such map becomes a quasi-homomorphism, and the paper shows that a previously published map advertised as a counterexample must have been built on an inaccurate calculation. The paper then extends the framework to higher degree by defining quasi-polynomial maps through bounded iterated non-commutative differences, and proves that a large class of quasi-quadratic maps is constructible in a precise sense. For torsion-free hyperbolic targets, it establishes a rigidity theorem: two nearly normal quasi-quadratic maps at finite distance are either both bounded, both map into the same cyclic subgroup, or are equal. This matters because it reduces a seemingly more flexible class of approximate homomorphisms to the known theory and provides a quadratic analogue of the rigidity quasi-homomorphisms enjoy in hyperbolic groups.

What carries the argument

For degree one, the carrying identity is $P_1(\varphi) = (D^*(\varphi_*))^{-1}$, where $\varphi_*(x)=\varphi(x)\varphi(1)^{-1}$ and $D^*$ is the right defect set $\{\varphi(x)\varphi(y)\varphi(xy)^{-1}\}$; boundedness of the middle defect of $\varphi$ is equivalent to boundedness of this right defect for $\varphi_*$. For degree two, the central object is the universal group $\mathrm{Pol}_2(G)$: the group generated by symbols $\tau(g)$ subject to the defining relations of quadratic maps, through which every unital quadratic map factors uniquely. The proof uses a finite-index subgroup of $\mathrm{Pol}_2(G)$ and the normality conditions (Q1)–(Q4), which require the image to be covered by finitely many cosets of the centralizer of the defect subgroup and the corresponding transversal to shrink; these conditions are what make the finite-index and finite-coset arguments work.

What would settle it

Compute the conjugation action in equation (4.9.4): with $[a,b]=z^\ell$ and $b^{-1}a^2=z^k$, the centralizer of $[a,b]$ is infinite cyclic and conjugation by $a$ is either trivial or inversion; the trivial case gives $\ell=0$, contradicting $[a,b]\neq 1$, while the inversion case gives $\ell=-2k$, not $k=-\ell$. A reader can check this in a free group or a torsion-free hyperbolic group, and the calculation shows the proof's asserted implication does not hold as written.

Watch

Extended reading notes

Core claim

The paper's central claim is that the middle defect condition carries no new degree of freedom: if the set $M(\varphi)=\{\varphi(x)^{-1}\varphi(xy)\varphi(y)^{-1}\mid x,y\in G\}$ is bounded, then $\varphi_*(x)=\varphi(x)\varphi(1)^{-1}$ is a quasi-homomorphism in the standard sense, and conversely every quasi-homomorphism right-multiplied by a constant has bounded middle defect. Hence every middle quasi-homomorphism is an affine quasi-homomorphism, and the paper's Theorem 1.1 classifies them completely. For the quadratic analogue, the paper introduces normal quasi-quadratic maps and shows they are constructible: on a finite-index subgroup of a universal group $\mathrm{Pol}_2(G)$, the map is within finite distance of a genuine quadratic map modulo a finitely generated central subgroup. The rigidity theorem for torsion-free hyperbolic targets then says that two nearly normal quasi-quadratic maps at finite distance coincide unless both are bounded or both have cyclic image.

Load-bearing premise

The proof of the quadratic rigidity theorem assumes, without derivation, that a certain relation between powers in a cyclic group forces two integer exponents to be negatives of each other; if that implication is false, the contradiction that yields rigidity collapses.

Editorial extensions

If this is right

Every middle quasi-homomorphism from an irreducible higher-rank lattice into a hyperbolic group, a mapping class group, or a fundamental group of a graph of hyperbolic groups is bounded, because the known rigidity of quasi-homomorphisms transfers directly.
For an unbounded quasi-homomorphism into a torsion-free hyperbolic group, the group $\Pi_\varphi$ of constants that preserve quasi-homomorphism status is either trivial or the unique cyclic subgroup containing the image; a non-trivial perturbing constant therefore forces the image to be cyclic.
Two nearly normal quasi-quadratic maps from the same group into a torsion-free hyperbolic group are rigid: if they are at finite distance, they are equal unless both are bounded or both land in a common cyclic subgroup.
Every normal quasi-quadratic map is constructible: up to bounded error and passage to a finite-index subgroup, it becomes a genuine quadratic map into a group with a finitely generated central kernel.

Reading between the lines

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

The same construction should work for quasi-polynomials of any degree: bounded iterated differences should force a decomposition into an exact polynomial map after finite-index restriction and central quotient, using higher-degree analogues of $\mathrm{Pol}_2(G)$.
The rigidity phenomenon for quasi-quadratic maps is likely to extend beyond hyperbolic groups to mapping class groups, CAT(0) groups, and groups acting on trees, mirroring the quasi-homomorphism results.
The multiplicative-quadruple criterion gives a practical boundedness test: to detect a middle quasi-homomorphism or a quasi-quadratic map one can sample the associated map on multiplicative quadruples rather than checking all pairs directly.

Share X Bluesky LinkedIn Reddit HN

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

You should know two things. Theorem 1.1 is the real news and it holds: middle quasi-homomorphisms are exactly constant perturbations of quasi-homomorphisms, and the proof is a short, clean consequence of Proposition 2.3. The paper also correctly identifies that this contradicts FK16 Theorem 9.6; the FK16 map, being unital, would have to be a quasi-homomorphism, and then Lemma 2.2 rules it out. That is a solid correction and an answer to an open question.

The quasi-polynomial framework is a genuine extension, not a routine one. The constructibility theorem (Theorem 1.3) is plausible and the multiplicative-quadruple criterion (Theorem 1.5) is a nice organizing principle. Proposition 4.7 is a clean hyperbolic-group consequence.

Now the soft spot. The stress-test note lands. In the proof of Theorem 1.4, right after (4.9.4), the paper asserts that if a does not centralize [a,b], then (4.9.4) implies k = -ℓ. It does not. In the only nontrivial case, conjugation by a inverts the cyclic centralizer of z, so (z^k)^a = z^{-k}, and (4.9.4) gives ℓ = -2k. The intersection C_H([a,b]) ∩ C_H([a,b])^a is then all of C_H([a,b]), so no contradiction follows from this line. The conclusion b = a^2, and hence the rigidity contradiction, depends entirely on k = -ℓ. The ℓ = -2k case may be excludable using the full Pol2Z relations or Lemma 4.6, but the paper does not say how. So Theorem 1.4 is unproved as written, even though I would bet it is true and fixable.

The rest of the paper does not seem to have a comparable problem. The main theorem and the framework stand independently of the rigidity theorem’s repair. For a reader working in geometric group theory or bounded cohomology, this is worth reading and citing for Theorem 1.1 and the quasi-polynomial definitions, with a citation note about Section 4. My recommendation: send it to peer review. A serious referee can push for the missing argument in Theorem 1.4 while the rest of the paper is publishable.

Referee Report

3 major / 4 minor

Summary. The paper studies maps between discrete groups with non-abelian targets, focusing on middle quasi-homomorphisms and quasi-polynomial maps. Its first main result, Theorem 1.1, asserts that middle quasi-homomorphisms are exactly constant perturbations of quasi-homomorphisms, contradicting a construction claimed in [FK16, Theorem 9.6]. The paper then characterizes the constants that can be used to perturb a quasi-homomorphism without leaving the quasi-homomorphism class (Theorem 1.2), develops a constructibility theory for quasi-quadratic maps via the group Pol2G (Theorem 1.3), and proves a rigidity theorem for nearly normal quasi-quadratic maps into torsion-free hyperbolic groups (Theorem 1.4).

Significance. If the results hold, Theorem 1.1 resolves an open question from Fujiwara--Kapovich and Theorem 1.4 extends rigidity from quasi-homomorphisms to quasi-quadratic maps, a promising direction connected to the polynomial-map framework of Jamneshan--Thom. The manuscript has several strengths: the main characterization is conceptually simple and likely correct after repair; the paper engages concretely with the prior literature; and the constructibility theorem provides a structured route from quasi-polynomial maps to quadratic maps modulo central subgroups. However, the current proofs contain a false auxiliary lemma and a load-bearing algebraic slip, and the rigidity proof has an unjustified step; these issues require repair before the claims can be accepted.

major comments (3)

[Section 2, Lemma 2.7] Lemma 2.7 is false as stated. Let G=H=Z with φ(n)=n+1 and ψ(n)=1. Then φ is a quasi-homomorphism with D(φ)={-1} and Δ_φ=Z, and ψ is a bounded map into Δ_φ, but γ(n)=ψ(n)φ(n)=n+2 is unbounded, contradicting the lemma's conclusion. The error occurs in the displayed computation where γ(y)=ψ(y)z_yt is replaced by ψ(y)t; although z_y centralizes Δ_φ, it is an unbounded factor and cannot be dropped. This lemma is invoked in Propositions 2.8, 3.1, 3.4 and 3.8, so all of these proofs need to be reworked.
[Section 2, Proposition 2.8] The proof of Proposition 2.8 contains an algebraically incorrect identity. The displayed chain asserts φ(x)^{-1}φ(xy)φ(y)^{-1} = (φ(y)^{-1}φ(x)^{-1}φ(xy))φ(y)^{-1}, which is not valid in a non-abelian group. The correct relation, using s=φ(y)^{-1}φ(x)^{-1}φ(xy)∈D(φ), is φ(x)^{-1}φ(xy)φ(y)^{-1}=φ(y)sφ(y)^{-1}. Since Corollary 2.10 and hence Theorem 1.1 depend on Proposition 2.8, this step is load-bearing. The statement of Proposition 2.8 appears to be repairable by combining Lemma 2.2(1) with the bounded-neighborhood decomposition φ(y)=z_y t_y, but the proof as written is not correct.
[Section 4, proof of Theorem 1.4, after equation (4.9.4)] The assertion that a∉C_H([a,b]) forces k=-ℓ from equation (4.9.4) is not justified by that equation alone. In the cyclic centralizer case, if a acts by the nontrivial automorphism z↦z^{-1}, then (4.9.4) becomes z^{ℓ+k}=z^{-k}, giving ℓ=-2k rather than k=-ℓ. To reach k=-ℓ one needs an additional argument, for example that in a torsion-free hyperbolic group an infinite intersection C_H([a,b])∩C_H([a,b])^a forces a∈C_H([a,b]), or a direct argument excluding the inversion automorphism. The subsequent conclusion b=a^2 and the final contradiction depend entirely on k=-ℓ, so this step must be supplied for Theorem 1.4 to be complete.

minor comments (4)

[Section 3, Proposition 3.7] The proof of Proposition 3.7 contains typos ('if Ci' should be 'if C_i') and a confusing step: the collection of cyclic subgroups containing φ(G) may be empty or poorly defined, and the sentence 'the intersection of the collection C contains an element k of infinite order' needs a clearer argument, especially when φ(G) is not itself contained in one cyclic subgroup.
[Section 2, after Corollary 2.10] The phrase 'slightly inaccurate calculation of the words2' appears to contain a typo and should be rewritten as 'of the words' or 'of the word calculations'.
[Section 3, Proposition 3.9] The statement of Proposition 3.9 has a typo: 'quasi-sungroup' should be 'quasi-subgroup'.
[Section 4, Proposition 4.4] The proof of Proposition 4.4 invokes Lemma 4.6 before it is stated; reordering the lemmas or adding a forward reference would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main characterization is derived directly from the definitions, and external citations are not used to force the conclusion.

full rationale

The paper's central result, Theorem 1.1, follows from elementary manipulations of the middle defect set: Proposition 2.5 shows right multiplication by a constant preserves being a middle quasi-homomorphism, and Corollary 2.4 shows a unital middle quasi-homomorphism is a quasi-homomorphism; composing these gives the characterization without assuming the conclusion. The contradiction with FK16 Theorem 9.6 is based on applying the external lemma FK16 Lemma 2.2 to Corollary 2.4, not on an imported uniqueness theorem. The constructibility results for quasi-quadratic maps use the external universal construction Pol2G from [JT24] as an ingredient, but the paper's representability claim is not the same as that construction and the argument does not reduce to it. No fitted parameters are present, and there are no relevant self-citations. The proof of Theorem 1.4 contains a potentially unproved algebraic step asserting that (4.9.4) forces k=-ell when a is not in C_H([a,b]); this is a mathematical-gap/correctness concern, not a circular dependence on the statement being proved. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard background theorems from the literature and on the standing assumption that H is discrete with a proper left-invariant metric.

assumptions (5)

domain assumption H is a discrete group with a proper left-invariant metric.
The paper assumes this setting throughout, inherited from FK16, so that boundedness of defect sets is meaningful. Section 1, first paragraph.
standard math Lemma 2.2 of FK16: for a quasi-homomorphism φ, φ(G) is contained in a bounded neighborhood of the centralizer of the defect subgroup Δ_φ.
Used in Lemma 2.7, Proposition 2.8, and the discussion of FK16 Theorem 9.6. Cited from [FK16, Lemma 2.3 and Corollary 3.1].
standard math JT24: The universal group Pol_2(G) exists, has an epimorphism π to G with abelian kernel, and every unital quadratic map factors uniquely through Pol_2(G).
Used in the proof of Theorem 1.3 via Lemma 4.6, which is cited from [JT24, Theorem 1.3].
standard math In a torsion-free hyperbolic group, the centralizer of a nontrivial element is cyclic, and the normalizer of an infinite cyclic subgroup equals its centralizer.
Used in Proposition 4.7 and Theorem 1.4, citing [BH99, Corollary III.Γ.3.10 and Proposition III.Γ.3.16].
standard math FK16 Theorem 4.4: unbounded quasi-homomorphisms into torsion-free hyperbolic groups are rigid under bounded perturbations in the sense that they map into the same cyclic subgroup.
Used in Proposition 3.7 and in the proof of Theorem 1.4 to handle the cyclic case.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quasi-affine and quasi-quadratic maps of groups with non-abelian targets." pith.science (2026). https://pith.science/paper/3BEZ7U6G

@misc{pith2026250601577,
  author       = {Pith},
  title        = {Pith review of: Quasi-affine and quasi-quadratic maps of groups with non-abelian targets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BEZ7U6G}},
  note         = {Machine review of arXiv:2506.01577}
}

read the original abstract

It is shown that the middle quasi-homomorphisms of Fujiwara and Kapovich are precisely constant perturbations of quasi-homomorphisms. Quasi-polynomial maps are defined and their constructibility is explored. In particular, it is shown that a large class of quasi-quadratic maps into torsion-free hyperbolic groups is rigid with respect to bounded perturbations.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

M. R. Bridson, and A. Haefliger, Metric spaces of non-positive curvature , Springer-Verlag, Berlin, Heidelberg, 1999

work page 1999
[2]

Brooks, Some remarks on bounded cohomology , in Annals of Mathematics Studies, Vol

R. Brooks, Some remarks on bounded cohomology , in Annals of Mathematics Studies, Vol. 97. Princeton University Press, Princeton, 1981

work page 1981
[3]

Cordes, T

M. Cordes, T. Hartnick, and V. Toni\' c , Foundations of geometric approximate group theory , arXiv:2012.15303v3, 1 Apr 2024. (with an appendix by S. Machado)

arXiv 2012
[4]

Fujiwara, and M

K. Fujiwara, and M. Kapovich, On quasihomomorphisms with noncommutative targets , Geom. Funct. Anal. 26 (2016), 478--519

work page 2016
[5]

Gowers, A new proof of Szemer\' edi's theorem for arithmetic progressions of length four , Geom

T. Gowers, A new proof of Szemer\' edi's theorem for arithmetic progressions of length four , Geom. Func. Anal. 8 (1998), 529-–551

work page 1998
[6]

Hartnick, and P

T. Hartnick, and P. Schweitzer, On quasioutomorphism groups of free groups and their transitivity properties , J. Algebra 450 (2016), 242--281

work page 2016
[7]

Heuer, Constructions in Stable Commutator Length and Bounded Cohomology , PhD thesis, 2019

N. Heuer, Constructions in Stable Commutator Length and Bounded Cohomology , PhD thesis, 2019

work page 2019
[8]

Hrushovski, Stable group theory and approximate subgroups , J

E. Hrushovski, Stable group theory and approximate subgroups , J. Amer. Math. Soc. 25 (1) (2012), 189--243

work page 2012

Show all 16 references

[9]

Jamneshan, and A

A. Jamneshan, and A. Thom, Quadratic maps between non-abelian groups , arXiv 2412.14908v1, 2024

2024
[10]

Leibman, Polynomial sequences in groups , J

A. Leibman, Polynomial sequences in groups , J. Algebra 201 (1998), no. 1, 189–-206

1998
[11]

Leibman, Polynomial mappings of groups , Israel J

A. Leibman, Polynomial mappings of groups , Israel J. Math. 129 (2002), 29–-60

2002
[13]

Machado, The structure of approximate lattices in linear groups , arXiv:2306.09899v2, 2023

S. Machado, The structure of approximate lattices in linear groups , arXiv:2306.09899v2, 2023

2023 arXiv
[14]

Tao, Product set estimates for non-commutative groups , Combinatorica 28 (2008), 547–-594

T. Tao, Product set estimates for non-commutative groups , Combinatorica 28 (2008), 547–-594

2008
[15]

Tao, and V

T. Tao, and V. Wu, Additive combinatorics , Cambridge University Press, 2007

2007
[16]

M. C. H. Tointon, Introduction to approximate groups , London Mathematical Society Student Texts 94, Cambridge University Press, 2020

2020
[17]

S. M. Ulam, A collection of mathematical problems , Interscience Tracts in Pure and Applied Mathematics, vol 8, Interscience Publishers,

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

M. R. Bridson, and A. Haefliger, Metric spaces of non-positive curvature , Springer-Verlag, Berlin, Heidelberg, 1999

work page 1999

[2] [2]

Brooks, Some remarks on bounded cohomology , in Annals of Mathematics Studies, Vol

R. Brooks, Some remarks on bounded cohomology , in Annals of Mathematics Studies, Vol. 97. Princeton University Press, Princeton, 1981

work page 1981

[3] [3]

Cordes, T

M. Cordes, T. Hartnick, and V. Toni\' c , Foundations of geometric approximate group theory , arXiv:2012.15303v3, 1 Apr 2024. (with an appendix by S. Machado)

arXiv 2012

[4] [4]

Fujiwara, and M

K. Fujiwara, and M. Kapovich, On quasihomomorphisms with noncommutative targets , Geom. Funct. Anal. 26 (2016), 478--519

work page 2016

[5] [5]

Gowers, A new proof of Szemer\' edi's theorem for arithmetic progressions of length four , Geom

T. Gowers, A new proof of Szemer\' edi's theorem for arithmetic progressions of length four , Geom. Func. Anal. 8 (1998), 529-–551

work page 1998

[6] [6]

Hartnick, and P

T. Hartnick, and P. Schweitzer, On quasioutomorphism groups of free groups and their transitivity properties , J. Algebra 450 (2016), 242--281

work page 2016

[7] [7]

Heuer, Constructions in Stable Commutator Length and Bounded Cohomology , PhD thesis, 2019

N. Heuer, Constructions in Stable Commutator Length and Bounded Cohomology , PhD thesis, 2019

work page 2019

[8] [8]

Hrushovski, Stable group theory and approximate subgroups , J

E. Hrushovski, Stable group theory and approximate subgroups , J. Amer. Math. Soc. 25 (1) (2012), 189--243

work page 2012