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arxiv: 2512.15375 · v2 · submitted 2025-12-17 · 🧮 math.GT · math.DS· math.GR

On quasimorphisms and distortion in homeomorphism groups

Michael Brandenbursky , Jarek Kedra , Michal Marcinkowski , Egor Shelukhin This is my paper

Pith reviewed 2026-05-16 21:36 UTC · model grok-4.3

classification 🧮 math.GT math.DSmath.GR MSC 57S05
keywords quasimorphismshomeomorphism groupsdiffeomorphism groupsdistortionbi-invariant metricsvolume preservingGambaudo-GhysPolterovich
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The pith

Gambaudo-Ghys and Polterovich quasimorphisms on diffeomorphism groups extend continuously to homeomorphism groups as quasimorphisms or cochains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies which members of two families of quasimorphisms defined on volume-preserving diffeomorphisms of a manifold extend continuously to the corresponding homeomorphism groups. These extensions are as quasimorphisms when preserving the measure and as group cochains with semi-bounded differentials when not. This connection matters because it allows the transfer of algebraic properties from the smooth category to the topological one. Readers interested in dynamics and group theory would care as it provides criteria for distortion and metric unboundedness in larger groups.

Core claim

We identify those Gambaudo-Ghys and Polterovich quasimorphisms Ψ: Diff_0(M,μ)→R which extend C^0-continuously to Homeo_0(M,μ) as quasimorphisms, and to Homeo_0(M) as group cochains whose differentials are semi-bounded cocycles. Several applications include the unboundedness of certain bi-invariant metrics on commutator subgroups and conditions for undistorted homeomorphisms.

What carries the argument

The Gambaudo-Ghys and Polterovich quasimorphisms, which are quasimorphisms on the diffeomorphism group that can be extended under continuity conditions to the homeomorphism group.

If this is right

  • The bi-invariant metric on the commutator subgroup of Homeo_0(M,μ) is unbounded.
  • Homeomorphisms satisfying certain conditions in Homeo_0(M) are undistorted.
  • The extensions preserve the quasimorphism property in the measure-preserving case.
  • Applications to detecting distortion in homeomorphism groups follow directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This suggests that continuity imposes strong restrictions, potentially limiting the variety of quasimorphisms on homeomorphism groups.
  • Similar extension results might apply to other classes of quasimorphisms or to higher regularity groups.
  • Testing on low-dimensional manifolds like the disk or sphere could verify the undistortion criteria explicitly.

Load-bearing premise

The result applies only to the Gambaudo-Ghys and Polterovich families of quasimorphisms rather than all possible quasimorphisms on these groups.

What would settle it

A concrete falsifier would be exhibiting one Gambaudo-Ghys quasimorphism on Diff_0(M,μ) that fails to extend continuously to Homeo_0(M,μ) while satisfying the other hypotheses.

Figures

Figures reproduced from arXiv: 2512.15375 by Egor Shelukhin, Jarek Kedra, Michael Brandenbursky, Michal Marcinkowski.

Figure 3.1
Figure 3.1. Figure 3.1: Isotopy inside ∆. Let x = (x1, . . . , xn). Assume first, that the evaluation {ft(xi)} of an isotopy {ft} is contained in ∆ for all i = 1, . . . , n. Then γ(f, x) ∈ Pn(∆). In this case β = γ(f, x) and α = e. A simplified picture is presented in [PITH_FULL_IMAGE:figures/full_fig_p009_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Back-and-forth. The green part is contained in Cn(⋃Bi). Moreover, since each Bi has radius less than 1 2n sys(Σ, d), every connected component of ⋃Bi has diameter less than sys(Σ) and is therefore contractible in Σ. Consequently, we can homotope the green part inside Cn(∆), keeping the endpoints fixed: (3.3) γ(f, x) = [ℓz,f(x) ∗ ℓf(x),y ∗ s ∗ ℓx,z] , where s denotes the green path after being pushed into… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Push into ∆ and back-and-forth again. each strand of α is a concatenation of three geodesic segments, each of length less than the diameter of Σ. The quasimorphism φ vanishes on Pn(∆) by hypothesis, hence the function (3.5) x ↦ ∣φ(γ(f, x))∣ = ∣φ(βα)∣ ≤ ∣φ(α)∣ + Dφ is bounded and so the integral Ψ(f) = ∫ Cn(Σ) φ(γ(f, x))dx is well defined, i.e. φ(γ(f, x)) is a L 1 -function, for a C 0 -small f. Let f ∈ Ho… view at source ↗
read the original abstract

Let $M$ be a smooth compact oriented connected manifold, and ${\rm Homeo}_0(M,\mu)$ the group of homeomorphisms of $M$ supported away from $\partial M,$ which preserve a Borel probability measure $\mu$ induced by a volume form on $M$, and are isotopic to the identity. In this paper, we identify those Gambaudo-Ghys and Polterovich quasimorphisms $\Psi\colon {\rm Diff}_0(M,\mu)\to R$ which extend $C^0$-continuously to ${\rm Homeo}_0(M,\mu)$ as quasimorphisms, and to ${\rm Homeo}_0(M)$ as group cochains whose differentials are semi-bounded cocycles. We present several applications of this result which include unboundedness of certain bi-invariant metric on the commutator subgroup of ${\rm Homeo}_0(M,\mu)$, and conditions under which a homeomorphism in ${\rm Homeo}_0(M)$ is undistorted.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper identifies those Gambaudo-Ghys and Polterovich quasimorphisms Ψ: Diff_0(M,μ) → ℝ that extend C⁰-continuously to Homeo_0(M,μ) as quasimorphisms and to Homeo_0(M) as group cochains whose differentials are semi-bounded cocycles. It derives applications to the unboundedness of certain bi-invariant metrics on the commutator subgroup of Homeo_0(M,μ) and to criteria for undistorted elements in Homeo_0(M).

Significance. If the extension criteria hold, the result supplies a concrete bridge between the smooth and topological categories for these specific families of quasimorphisms, yielding direct consequences for bi-invariant metrics and distortion phenomena on homeomorphism groups of manifolds. The explicit restriction to the Gambaudo-Ghys and Polterovich families avoids overclaiming a full classification.

minor comments (3)
  1. §1 (Introduction): the precise definition of 'semi-bounded cocycle' should be recalled or referenced immediately after its first use to aid readers unfamiliar with the Polterovich construction.
  2. §4 (Applications): the bi-invariant metric whose unboundedness is claimed should be named explicitly (e.g., the fragmentation metric or the one induced by the quasimorphism) rather than referred to generically.
  3. Notation: the distinction between Homeo_0(M,μ) and Homeo_0(M) is clear in the abstract but could be reinforced with a short table or diagram in §2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and the recommendation for minor revision. The report does not list any specific major comments, so we have no points to address point-by-point. We will proceed with minor revisions to improve clarity and presentation as appropriate.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper restricts attention to the Gambaudo-Ghys and Polterovich families of quasimorphisms on Diff_0(M,μ) and derives criteria for their C^0-continuous extension to Homeo_0(M,μ) as quasimorphisms and to Homeo_0(M) as cochains with semi-bounded differentials. These criteria are established by direct analysis of the quasimorphism properties and continuity conditions rather than by fitting parameters to data, renaming known results, or reducing the central claim to a self-citation chain. The applications to bi-invariant metrics and undistorted elements follow directly from the extension criteria once proven for the specified families. No load-bearing step reduces by construction to the paper's own inputs or prior fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard definitions of quasimorphisms, C^0 topology, and bi-invariant metrics on homeomorphism groups; no new free parameters, ad-hoc axioms, or invented entities are introduced in the visible statement.

axioms (2)
  • standard math Quasimorphisms are maps satisfying the bounded defect inequality |ψ(gh) - ψ(g) - ψ(h)| ≤ D for some constant D.
    Standard definition used throughout the field and presupposed by the statement.
  • domain assumption The groups Diff_0(M,μ) and Homeo_0(M,μ) are equipped with the C^0 topology induced by the manifold structure.
    Background topology on the groups of maps.

pith-pipeline@v0.9.0 · 5486 in / 1363 out tokens · 51026 ms · 2026-05-16T21:36:34.519615+00:00 · methodology

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

  1. [1]

    Bounded cohomology of subgroups of mapping class groups.Geom

    Mladen Bestvina and Koji Fujiwara. Bounded cohomology of subgroups of mapping class groups.Geom. Topol., 6:69–89, 2002

    work page 2002

  2. [2]

    Joan S. Birman. On braid groups.Comm. Pure Appl. Math., 22:41–72, 1969

    work page 1969

  3. [3]

    Quasi-morphisms on surface diffeomorphism groups.J

    Jonathan Bowden, Sebastian Wolfgang Hensel, and Richard Webb. Quasi-morphisms on surface diffeomorphism groups.J. Am. Math. Soc., 35(1):211–231, 2022

    work page 2022

  4. [4]

    On quasi-morphisms from knot and braid invariants.J

    Michael Brandenbursky. On quasi-morphisms from knot and braid invariants.J. Knot Theory Ramifications, 20(10):1397–1417, 2011

    work page 2011

  5. [5]

    Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces.Int

    Michael Brandenbursky. Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces.Int. J. Math., 26(9):29, 2015. Id/No 1550066

    work page 2015

  6. [6]

    Gal, Jarek K¸ edra, and Michał Marcinkowski

    Michael Brandenbursky, Światosław R. Gal, Jarek K¸ edra, and Michał Marcinkowski. The cancellation norm and the geometry of bi-invariant word metrics.Glasg. Math. J., 58(1):153–176, 2016

    work page 2016

  7. [7]

    On the autonomous metric on the group of area-preserving diffeomor- phisms of the 2-disc.Algebr

    Michael Brandenbursky and Jarek K¸ edra. On the autonomous metric on the group of area-preserving diffeomor- phisms of the 2-disc.Algebr. Geom. Topol., 13(2):795–816, 2013

    work page 2013

  8. [8]

    Fragmentation norm and relative quasimorphisms.Proc

    Michael Brandenbursky and Jarek Kędra. Fragmentation norm and relative quasimorphisms.Proc. Am. Math. Soc., 150(10):4519–4531, 2022

    work page 2022

  9. [9]

    MichaelBrandenbursky, JarekKędra, andEgorShelukhin.OntheautonomousnormonthegroupofHamiltonian diffeomorphisms of the torus.Commun. Contemp. Math., 20(2):27, 2018. Id/No 1750042

    work page 2018

  10. [10]

    Entropy and quasimorphisms.J

    Michael Brandenbursky and Michał Marcinkowski. Entropy and quasimorphisms.J. Mod. Dyn., 15:143–163, 2019

    work page 2019

  11. [11]

    Ann., 382(3-4):1181–1197, 2022

    MichaelBrandenburskyandMichałMarcinkowski.Boundedcohomologyoftransformationgroups.Math. Ann., 382(3-4):1181–1197, 2022

    work page 2022

  12. [12]

    The $L^p$-diameter of the space of contractible loops

    Michael Brandenbursky and Egor Shelukhin. TheL p-diameter of the space of contractible loops. Preprint, arXiv:2509.07270, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  13. [13]

    Danny Calegari.scl., volume 20 ofMSJ Mem.Tokyo: Mathematical Society of Japan, 2009

    work page 2009

  14. [14]

    Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

    Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak, Sobhan Seyfaddini, and Ivan Smith. Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces. Preprint, arXiv:2206.10749, 2025

    work page arXiv 2025

  15. [15]

    R. D. Edwards and R. C. Kirby. Deformations of spaces of imbeddings.Ann. Math. (2), 93:63–88, 1971

    work page 1971

  16. [16]

    A two-cocycle on the group of symplectic diffeomorphisms.Math

    Światosław Gal and Jarek Kędra. A two-cocycle on the group of symplectic diffeomorphisms.Math. Z., 271(3- 4):693–706, 2012

    work page 2012

  17. [17]

    ŚwiatosławR.GalandJarekKędra.Arefinementofboundedcohomology.Preprint, arXiv:2208.03168[math.GR] (2022), 2022

    work page arXiv 2022

  18. [18]

    Gambaudo and E

    J.-M. Gambaudo and E. E. Pécou. Dynamical cocycles with values in the Artin braid group.Ergodic Theory Dynam. Systems, 19(3):627–641, 1999

    work page 1999

  19. [19]

    Commutators and diffeomorphisms of surfaces.Ergodic Theory Dyn

    Jean-Marc Gambaudo and Étienne Ghys. Commutators and diffeomorphisms of surfaces.Ergodic Theory Dyn. Syst., 24(5):1591–1617, 2004

    work page 2004

  20. [20]

    Goldberg

    Charles H. Goldberg. An exact sequence of braid groups.Math. Scand., 33:69–82, 1973

    work page 1973

  21. [21]

    Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups.Proc

    Tomohiko Ishida. Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups.Proc. Am. Math. Soc., Ser. B, 1:43–51, 2014

    work page 2014

  22. [22]

    Ivanov.Subgroups of Teichmüller modular groups, volume 115 ofTranslations of Mathe- matical Monographs

    Nikolai V. Ivanov.Subgroups of Teichmüller modular groups, volume 115 ofTranslations of Mathe- matical Monographs. American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author

    work page 1992

  23. [23]

    Kolev and M.-C

    B. Kolev and M.-C. Pérouème. Recurrent surface homeomorphisms.Math. Proc. Camb. Philos. Soc., 124(1):161–168, 1998

    work page 1998

  24. [24]

    Distortion elements of Diff∞ 0 (M).Bull

    Emmanuel Militon. Distortion elements of Diff∞ 0 (M).Bull. Soc. Math. Fr., 141(1):35–46, 2013. 15

    work page 2013

  25. [25]

    Distortion elements for surface homeomorphisms.Geom

    Emmanuel Militon. Distortion elements for surface homeomorphisms.Geom. Topol., 18(1):521–614, 2014

    work page 2014

  26. [26]

    Floer homology, dynamics and groups

    Leonid Polterovich. Floer homology, dynamics and groups. InMorse theoretic methods in nonlinear anal- ysis and in symplectic topology. Proceedings of the NATO Advanced Study Institute, Montréal, Canada, July 2004, pages 417–438. Dordrecht: Springer, 2006

    work page 2004

  27. [27]

    Quasi-morphismes de Calabi et graphe de Reeb sur le tore.C

    Pierre Py. Quasi-morphismes de Calabi et graphe de Reeb sur le tore.C. R. Math. Acad. Sci. Paris, 343(5):323–328, 2006

    work page 2006

  28. [28]

    Alexander I. Shtern. Remarks on pseudocharacters and the real continuous bounded cohomology of connected locally compact groups.Ann. Global Anal. Geom., 20(3):199–221, 2001

    work page 2001

  29. [29]

    A remark on the topological entropy of homeomorphisms.Invent

    Koichi Yano. A remark on the topological entropy of homeomorphisms.Invent. Math., 59:215–220, 1980. Department of Mathematics, Ben Gurion University, Israel Email address:brandens@bgu.ac.il Department of Mathematics, University of Aberdeen, UK and University of Szczecin, Poland Email address:kedra@abdn.ac.uk Department of Mathematics, University of Wrocła...

    work page 1980