Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

Lior Menashe; Michael Brandenbursky

REVIEW 1 major objections 2 minor 15 references

Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces

T0 review · 1 major / 2 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The identity component of the measure-preserving homeomorphism group of every closed non-orientable surface of genus at least three has infinite-dimensional bounded cohomology in degrees two and three.

desk verdict The H^3 result is likely real, but the proof has a load-bearing gap in the hyperbolic embedding claim for g=3. read the letter →

arxiv 2506.02728 v1 pith:JNB7ZCHO submitted 2025-06-03 math.GT math.GR

Michael Brandenbursky , Lior Menashe This is my paper

classification math.GTmath.GR MSC 57M6020J0657S05

keywords boundedcohomologymeasure-preservinghomeomorphismgroupsnon-orientablesurfaceshyperbolicallyembeddedsubgroupsGambaudo-Ghysmapexactsurfacegroup

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 proves that for every closed non-orientable surface $N_g$ of genus $g\geq 3$, the identity component $\operatorname{Homeo}_0(N_g,\mu)$ of its measure-preserving homeomorphism group carries infinitely many linearly independent bounded cohomology classes in degrees 2 and 3, so both $H^2_b(\operatorname{Homeo}_0(N_g,\mu))$ and $H^3_b(\operatorname{Homeo}_0(N_g,\mu))$ are infinite-dimensional. This fills the non-orientable gap left by recent work on orientable manifolds. The proof builds a Gambaudo–Ghys map that integrates bounded classes of the surface group over the surface, and shows that on exact bounded cohomology it is injective on an infinite-dimensional subspace. The decisive step for the difficult genera $g=3,4$ is a new proof that the free group generated by two loops is hyperbolically embedded in the non-orientable surface group, which lets a known surjectivity theorem transfer infinite dimensionality from the two-generator free group to the homeomorphism group. A reader should care because transformation-group bounded cohomology is a dynamical and geometric invariant, and the result completes the low-degree picture for closed surfaces.

What carries the argument

The workhorse is the Gambaudo–Ghys map, defined here as $\Gamma_b^n:H_b^n(\pi_1(N_g))\to H_b^n(\operatorname{Homeo}_0(N_g,\mu))$, which sends a bounded cocycle $c$ to the class of $(g_0,\dots,g_n)\mapsto \int_{N_g} c(\gamma(g_0,x),\dots,\gamma(g_n,x))\,d\mu(x)$, where $\gamma(g,x)$ is the element of $\pi_1(N_g)$ traced out by the point $x$ during an isotopy from the identity to $g$. The load-bearing new input is Proposition 4.4: it proves that $F_2=\langle a,b\rangle$ is hyperbolically embedded in $\pi_1(N_g)$ for every $g\geq 3$. The proof of the difficult condition (c), that every finite-radius ball in the auxiliary metric $\hat d$ on $F_2$ is finite, is an inductive description of admissible geodesics: in the genus-3 case every admissible geodesic from $e$ to an element of $F_2$ has the form $e,c,c^2,\dots$ or $e,c^{-1},c^{-2},\dots$, which makes the balls $\{(aba^{-1}b^{-1})^l: |l|<r\}$ finite. That finite-ball condition is exactly what lets the restriction theorem for hyperbolically embedded subgroups apply and yields the dimension inequality.

What would settle it

In the genus-3 presentation $\langle a,b,c\mid aba^{-1}b^{-1}c^2=1\rangle$ with the added generator set $X=\{c\}$, compute the auxiliary distance $\hat d(e,h)$ for every reduced word $h$ in $a,b$ up to increasing length, using paths that contain no step inside the subgroup generated by $a,b$. If any such shortest path differs from the claimed patterns $e,c,c^2,\ldots$ or $e,c^{-1},c^{-2},\ldots$, or if infinitely many distinct $h$ have $\hat d(e,h)$ bounded by one fixed constant, then the finite-ball condition fails, the free group is not hyperbolically embedded in the required way, and the surjectivity step in the proof collapses.

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Extended reading notes

Core claim

The central claim is Theorem 1: for $g\geq 3$, both $\dim H^2_b(\operatorname{Homeo}_0(N_g,\mu))$ and $\dim H^3_b(\operatorname{Homeo}_0(N_g,\mu))$ are infinite, and the proof actually gives the continuum cardinal $2^{\aleph_0}$. For $g\geq 5$ the argument is an extension of the orientable one: a figure-eight embedded in $N_g$ admits a retraction of the whole surface onto it, so restriction and projection maps on exact reduced bounded cohomology give the required transfer. For $g=3,4$ no such retraction exists, and the authors prove instead that $F_2=\langle a,b\rangle$ is hyperbolically embedded in $\pi_1(N_g)$ by controlling all admissible geodesics in a Cayley graph with the added relator letter. A known theorem on hyperbolically embedded subgroups then makes the restriction map from $\pi_1(N_g)$ to $F_2$ surjective on exact reduced bounded cohomology, and a family of measure-preserving maps $\rho_\epsilon:F_2\to\operatorname{Homeo}_0(N_g,\mu)$ approximating the projection transfers the known uncountably many classes of the two-generator free group to classes of $\operatorname{Homeo}_0(N_g,\mu)$.

Load-bearing premise

The entire argument rests on showing that the free subgroup generated by the two figure-eight loops is isolated inside the non-orientable surface group in a strong sense: when distance is measured by paths that may not shortcut through that subgroup, every ball of bounded radius contains only finitely many elements of the subgroup.

Editorial extensions

If this is right

For every $g\geq 3$, $\operatorname{Homeo}_0(N_g,\mu)$ has at least $2^{\aleph_0}$ linearly independent exact bounded classes in degree 2 and in degree 3.
The non-orientable cases $g=3,4$ are covered even though the surface group does not retract onto the free subgroup; the hyperbolic-embedding argument is what replaces the missing retraction.
The Gambaudo–Ghys map is nontrivial on an infinite-dimensional subspace for degrees 2 and 3, so integration over the surface detects genuinely new bounded classes of transformation groups.
For $g=1,2$ the method stops, because the fundamental groups of those non-orientable surfaces have trivial exact bounded cohomology; the paper identifies those cases as requiring new ideas.

Reading between the lines

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

The same hyperbolic-embedding-and-approximation scheme should work for compact non-orientable surfaces with boundary or with punctures, where the fundamental group is free and the low-degree exact bounded cohomology is already known to be infinite-dimensional.
The proof separates the topological input, hyperbolic embedding, from the analytic input, the approximation by measure-preserving rotations; this suggests the infinite-dimensionality is robust under changing the smoothness class from homeomorphisms to diffeomorphisms as long as the same measure class is preserved.
A testable consequence of the argument is that the constructed bounded classes lie in the kernel of the comparison map to ordinary cohomology, so they cannot be detected by characteristic classes coming from the ordinary cohomology of the homeomorphism group.
For the remaining genera $g=1,2$, the obstruction is not merely technical: since the surface group has no exact bounded classes, a genuinely different source of bounded classes would be needed if the same conclusion still holds there.

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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

The new result here is honest progress: for every closed non-orientable surface of genus at least 3, the third bounded cohomology of the identity component of measure-preserving homeomorphisms is infinite dimensional. That completes the non-orientable analogue of known orientable results, and H^2 was already in Kumar's thesis. The overall strategy is sound in outline: transfer map, hyperbolic embedding of F2, and the Gambaudo–Ghys integration against the measure. The explicit constructions for g=4 and g≥5 look credible, and the authors are careful to credit prior work. This is not a fake or inflated paper.

But there is a real problem in the g=3 case, and it is not just a sketch. In Proposition 4.4 the authors claim that H = ⟨a,b⟩ is hyperbolically embedded in G = ⟨a,b,c | aba⁻¹b⁻¹c² = 1⟩. The stress-test note is right: the relation gives c² = ba b⁻¹a⁻¹ ∈ H, and since c ∉ H (the abelianization shows c has order 2 while a,b are torsion-free), the intersection H ∩ cHc⁻¹ contains the infinite cyclic subgroup ⟨c²⟩. So H is not almost malnormal, and Proposition 2.4 — which they themselves cite — says a hyperbolically embedded subgroup must be almost malnormal. That is a genuine contradiction. The inductive description of admissible geodesics in the proof of condition (c) cannot be right: the path e, c, c² already shows that the alleged ‘only two edges’ claim fails. The paper's own Remark 4.6 even discusses a similar non-example, which makes the oversight more puzzling. The final argument for all g≥3 invokes Proposition 4.4, so this gap undermines the proof as written. The theorem may still be true — the g=4 and g≥5 cases are independent and likely fine — but the g=3 case is not established.

There are also minor issues: the final paragraph swaps injection and surjection for i^n and π^n, and the proof of Proposition 4.4 condition (b) for g=3 is only sketched. Those are repairable. The swapped sentence is cosmetic. The hyperbolic embedding failure is structural.

Who is this for? Researchers in bounded cohomology of transformation groups and geometric group theorists working with hyperbolically embedded subgroups. The paper deserves serious referee attention — the question is genuine and the g≥5 argument is solid — but only if the authors can fix or replace the g=3 embedding. I would not cite the main theorem yet. I would send to a journal with a referee who knows hyperbolic embeddings, not just bounded cohomology.

Referee Report

1 major / 2 minor

Summary. The paper defines a Gambaudo-Ghys transfer map for the identity component of the group of measure-preserving homeomorphisms of a closed non-orientable surface and uses it, together with a claimed hyperbolic embedding of F2 into π1(N_g), to prove that H^2_b and H^3_b of Homeo_0(N_g, μ) are infinite-dimensional for every g ≥ 3. The proof has two parts: an elementary retraction argument for g ≥ 5, and a general argument via Proposition 4.4, which asserts that a particular F2 subgroup of π1(N_g) is hyperbolically embedded for every g ≥ 3.

Significance. If the theorem were correct, it would be a natural non-orientable analogue of recent results on bounded cohomology of transformation groups and would supply the first higher bounded classes for measure-preserving homeomorphism groups of non-orientable surfaces. The Gambaudo-Ghys transfer construction and the Moser-theorem support for the maps ρ_ε are well motivated. However, the main theorem depends on Proposition 4.4, and that proposition is false; the submitted proof therefore does not establish the stated result.

major comments (1)

[Section 4.2, Proposition 4.4, Case 1] The hyperbolic embedding claim is false. In G = ⟨a,b,c | a b a^{-1} b^{-1} c^2 = 1⟩, the relation gives c^2 = b a b^{-1} a^{-1} ∈ H. For every n ≥ 2, the path e, c, c^{2n-1}, c^{2n} in Γ(G, {c} ⊔ H) is admissible: the middle edge is labeled by c^{2n-2} ∈ H and the other two are labeled by c, and none of the intermediate vertices lies in H. Hence d_hat(e, c^{2n}) ≤ 3 for infinitely many distinct elements c^{2n} ∈ H, so the ball B_dhat(e, 3) is infinite. This directly contradicts condition (c) of Definition 2.1. Equivalently, H is not almost malnormal: H ∩ cHc^{-1} contains the infinite cyclic subgroup ⟨c^2⟩, since c ∉ H (visible in the abelianization G_ab ≅ Z^2 ⊕ Z/2, where c has order 2 and a, b are free) and c c^2 c^{-1} = c^2. Proposition 2.4 then rules out hyperbolic embeddedness. The proof of condition (c) in the paper claims that the only admissible geodesics are e, c, c^2, ... or e, c^{-1}, c^{-2}, ..., which is false; the path e, c, c^{2n-1}, c^{2n} is a counterexample. Consequently Corollary 4.5 and the dimension bound for g = 3 are not established, and the 'general case' argument for all g ≥ 3, which invokes Proposition 4.4, collapses.

minor comments (2)

[Section 4, final paragraph] The sentence 'whenever g ≥ 5 the map i^n is an injection, and π^n is a surjection' reverses the statements of Corollary 4.2; Corollary 4.2 gives i^n surjective and π^n injective. The subsequent composition argument is repairable, but as written it is inconsistent with the earlier corollary.
[Remark 1.1 and Section 2.2] Remark 1.1 says that the proof of the H^2 statement is different from Kumar's, but Section 2.2 invokes Kumar's thesis for exactly this statement; please clarify which proof is meant.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependency found; the main derivation rests on external results and in-paper constructions.

full rationale

The paper's derivation chain is: (i) external results of Brooks and Soma give infinite-dimensional exact bounded cohomology of F2; (ii) the external Frigerio--Pozzetti--Sisto surjectivity theorem is applied to a claimed hyperbolic embedding; (iii) Proposition 4.4 tries to verify the embedding by constructing admissible geodesics; (iv) Lemma 4.7 constructs the maps rho_epsilon explicitly using Moser's theorem, and the convergence argument is carried out in the text. The only self-citations are methodological, most prominently 'We proceed as in [3]' in Sections 4.2 and 4.3, and the cited construction is re-proved in Lemma 4.7 rather than assumed. No equation is defined in terms of the target claim, no fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation chain. The suspected flaw in Proposition 4.4, that H = <a,b> is not almost malnormal because c^2 lies in H cap cHc^{-1}, is a mathematical correctness issue, not a circularity: a false lemma still does not make the argument's output equal to its input by definition. Therefore the circularity score is 0.

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

The argument is built from standard or cited theorems plus a new hyperbolic embedding proof. No fitted constants or new entities are introduced. Lambda is a geometric constant depending on chosen neighborhoods, nonzero by construction, and no numerical value is needed for the proof.

assumptions (5)

domain assumption Brooks [5] and Soma [15]: dim EH^n_b(F2) = 2^{aleph0} for n=2,3.
Provides the nonzero classes that are transported to Homeo_0(N_g,mu); cited in Section 4, not proved.
domain assumption Frigerio-Pozzetti-Sisto [7, Corollary 2]: restriction map EH^n_b(G) to EH^n_b(H) is surjective for hyperbolically embedded H, n≥2.
Used to conclude i^n is surjective in Corollary 4.5; cited as Theorem 4.3.
domain assumption Center of π1(N_g) is trivial for g≥3, so the evaluation map ev_z^* has trivial image.
Needed in Proposition 3.1 to prove γ is independent of isotopy; stated as 'which is trivial' without proof.
standard math Moser's theorem supplies a measure-preserving diffeomorphism between the tubular neighborhood and the annulus.
Used in Lemma 4.7 to construct area-preserving isotopies; cited generically.
domain assumption EH^n_b(Z)=0.
Used in Lemma 4.7 to kill contributions from A_a^epsilon and A_b^epsilon.

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Pith. "Pith review of Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces." pith.science (2026). https://pith.science/paper/JNB7ZCHO

@misc{pith2026250602728,
  author       = {Pith},
  title        = {Pith review of: Bounded cohomology of measure-preserving homeomorphism groups of non-orientable surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JNB7ZCHO}},
  note         = {Machine review of arXiv:2506.02728}
}

abstract

Let $N_g$ be a closed non-orientable surface of genus $g\geq 3$. Let $\operatorname{Homeo}_0(N_g,\mu)$ be the identity component of the group of measure-preserving homeomorphisms of $N_g$. In this work we prove that the third bounded cohomology of $\operatorname{Homeo}_0(N_g,\mu)$ is infinite dimensional.

Figures

Figures reproduced from arXiv: 2506.02728 by the authors.

**Figure 1.** i : S 1 ∨ S 1 ,→ N5 and π : N5 ↠ S 1 ∨ S 1 The image of the map i : S 1 ∨ S 1 ,→ N5 is the blue figure-eight, and the map π : N5 ↠ S 1 ∨ S 1 is an obvious projection. The case g > 5 follows immediately since Ng ∼= T 2#T 2# RP 2 # . . . #RP 2 | {z } g − 4 times . □ Corollary 4.2. On the level of the exact reduced bounded cohomology i n ◦ π n = Id, therefore i n is onto and π n is injective. 4.2. General case. In orde… view at source ↗

**Figure 2.** i : S 1 ∨ S 1 ,→ N3 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The geodesic in Γ from e to h We proceed with the path e, c, c2 inductively: an element c 2n belongs to H, hence the only two edges staring at c 2n and are not contained in ΓH are labeled by c 2n+1 and c 2n−1 . A geodesic that ends in H can not start in an edge which belongs to the orbit c 2n+1ΓH. Therefore, the only two edges which start at c 2n+1, such that the resulting geodesic ends in H are again labeled by c a… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The admissible path e, a, a2n−1 , a2n when H = ⟨a 2 , b2 ⟩ 4.3. Construction of ρϵ : F2 → Homeo0(Ng, µ). Let i be an embedding of figure-eight discussed in the previous section, i.e., i n is surjective. Now we proceed as in [3] [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: The loop α is free homotopic (or conjugated in π1(Ng, z)) to the loop γ(ρϵ(a), x) whenever x ∈ N(α). Let ω ∈ F2. By using Proposition 3.2 we get: γ(ρϵ(ω), x) =    [e], x ∈ Ng\(N(α) ∪ N(β)) [ux][ω][ux] −1 , x ∈ Aϵ = Aϵ(α) ∩ Aϵ(β) [ux,a][ha(ω)][ux,a] −1 , x ∈ Aa…

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Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Springer, Dordrecht, 2006

Paul Biran, Octav Cornea, and Fran¸cois Lalonde, editors.Morse theoretic methods in nonlinear analysis and in symplectic topology, volume 217 ofNATO Science Series II: Mathematics, Physics and Chemistry. Springer, Dordrecht, 2006. 4

work page 2006
[2]

Entropy and quasimorphisms.J

Michael Brandenbursky and MichałMarcinkowski. Entropy and quasimorphisms.J. Mod. Dyn., 15:143–163, 2019. 1

work page 2019
[3]

Bounded cohomology of transfor- mation groups.Math

Michael Brandenbursky and Michal Marcinkowski. Bounded cohomology of transfor- mation groups.Math. Ann., 382(3-4):1181–1197, 2022. 1, 4, 6, 10

work page 2022
[4]

Volume and Euler classes in bounded cohomology of transformation groups.Glasg

Michael Brandenbursky and MichałMarcinkowski. Volume and Euler classes in bounded cohomology of transformation groups.Glasg. Math. J., 67(1):34–49, 2025. 5

work page 2025
[5]

Some remarks on bounded cohomology

Robert Brooks. Some remarks on bounded cohomology. InRiemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume No. 97 ofAnn. of Math. Stud., pages 53–63. Princeton Univ. Press, Princeton, NJ, 1981. 6

work page 1978
[6]

Dahmani, V

F. Dahmani, V. Guirardel, and D. Osin. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces.Mem. Amer. Math. Soc., 245(1156):v+152, 2017. 3

work page 2017
[7]

Frigerio, M

R. Frigerio, M. B. Pozzetti, and A. Sisto. Extending higher-dimensional quasi- cocycles.J. Topol., 8(4):1123–1155, 2015. 7

work page 2015
[8]

Commutators and diffeomorphisms of sur- faces.Ergodic Theory Dynam

Jean-Marc Gambaudo and Étienne Ghys. Commutators and diffeomorphisms of sur- faces.Ergodic Theory Dynam. Systems, 24(5):1591–1617, 2004. 4

work page 2004

Show all 15 references

[9]

Volume and bounded cohomology.Inst

Michael Gromov. Volume and bounded cohomology.Inst. Hautes Études Sci. Publ. Math., (56):5–99, 1982. 1

1982
[10]

ˆG-invariant quasimorphisms and sym- plectic geometry of surfaces.Israel J

Morimichi Kawasaki and Mitsuaki Kimura. ˆG-invariant quasimorphisms and sym- plectic geometry of surfaces.Israel J. Math., 247(2):845–871, 2022. 1

2022
[11]

Gambaudo-Ghys construction on bounded cohomology.J

Mitsuaki Kimura. Gambaudo-Ghys construction on bounded cohomology.J. Math. Soc. Japan, 77(1):135–152, 2025. 1

2025
[12]

Construction of quasi-morphisms on groups of measure-preserving homeomorphisms of non-orientable surfaces.MSc

Rishi Kumar. Construction of quasi-morphisms on groups of measure-preserving homeomorphisms of non-orientable surfaces.MSc. thesis, Ben Gurion University of the Negev, 2021. 2, 3

2021
[13]

Higher-degree bounded cohomology of transformation groups

Martin Nitsche. Higher-degree bounded cohomology of transformation groups. arXiv:2105.08698, 2021. 1, 5

2021 arXiv
[14]

D. Osin. Acylindrically hyperbolic groups.Trans. Amer. Math. Soc., 368(2):851–888,
[15]

Bounded cohomology and topologically tame Kleinian groups.Duke Math

Teruhiko Soma. Bounded cohomology and topologically tame Kleinian groups.Duke Math. J., 88(2):357–370, 1997. 6 Department of Mathematics, Ben Gurion University, Israel Email address:brandens@bgu.ac.il Department of Mathematics, Ben Gurion University, Israel Email address:lior....

1997

Pith tools

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

[1] [1]

Springer, Dordrecht, 2006

Paul Biran, Octav Cornea, and Fran¸cois Lalonde, editors.Morse theoretic methods in nonlinear analysis and in symplectic topology, volume 217 ofNATO Science Series II: Mathematics, Physics and Chemistry. Springer, Dordrecht, 2006. 4

work page 2006

[2] [2]

Entropy and quasimorphisms.J

Michael Brandenbursky and MichałMarcinkowski. Entropy and quasimorphisms.J. Mod. Dyn., 15:143–163, 2019. 1

work page 2019

[3] [3]

Bounded cohomology of transfor- mation groups.Math

Michael Brandenbursky and Michal Marcinkowski. Bounded cohomology of transfor- mation groups.Math. Ann., 382(3-4):1181–1197, 2022. 1, 4, 6, 10

work page 2022

[4] [4]

Volume and Euler classes in bounded cohomology of transformation groups.Glasg

Michael Brandenbursky and MichałMarcinkowski. Volume and Euler classes in bounded cohomology of transformation groups.Glasg. Math. J., 67(1):34–49, 2025. 5

work page 2025

[5] [5]

Some remarks on bounded cohomology

Robert Brooks. Some remarks on bounded cohomology. InRiemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume No. 97 ofAnn. of Math. Stud., pages 53–63. Princeton Univ. Press, Princeton, NJ, 1981. 6

work page 1978

[6] [6]

Dahmani, V

F. Dahmani, V. Guirardel, and D. Osin. Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces.Mem. Amer. Math. Soc., 245(1156):v+152, 2017. 3

work page 2017

[7] [7]

Frigerio, M

R. Frigerio, M. B. Pozzetti, and A. Sisto. Extending higher-dimensional quasi- cocycles.J. Topol., 8(4):1123–1155, 2015. 7

work page 2015

[8] [8]

Commutators and diffeomorphisms of sur- faces.Ergodic Theory Dynam

Jean-Marc Gambaudo and Étienne Ghys. Commutators and diffeomorphisms of sur- faces.Ergodic Theory Dynam. Systems, 24(5):1591–1617, 2004. 4

work page 2004