Geodesic
The space of nonextendable quasimorphisms
Kawasaki, Morimichi1 ; Kimura, Mitsuaki2 ; Maruyama, Shuhei3 ; Matsushita, Takahiro4 ; Mimura, Masato5
1Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan
2Department of Mathematics, Osaka Dental University, Hirakata, Japan
3School of Mathematics and Physics, College of Science and Engineering, Kanazawa University, Kanazawa, Japan
4Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Japan
5Mathematical Institute, Tohoku University, Sendai, Japan
Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 1169-1226
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

For a pair (G,N) of a group G with normal subgroup N, we consider the space of quasimorphisms and quasicocycles on N nonextendable to G. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As an application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov-hyperbolic groups.

Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and normal subgroup.

DOI : 10.2140/agt.2025.25.1169
Keywords: bounded cohomology, invariant quasimorphisms, group cohomology, bounded acyclicity

Kawasaki, Morimichi  1   ; Kimura, Mitsuaki  2   ; Maruyama, Shuhei  3   ; Matsushita, Takahiro  4   ; Mimura, Masato  5

1 Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan
2 Department of Mathematics, Osaka Dental University, Hirakata, Japan
3 School of Mathematics and Physics, College of Science and Engineering, Kanazawa University, Kanazawa, Japan
4 Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Japan
5 Mathematical Institute, Tohoku University, Sendai, Japan 
@article{10_2140_agt_2025_25_1169,
     author = {Kawasaki, Morimichi and Kimura, Mitsuaki and Maruyama, Shuhei and Matsushita, Takahiro and Mimura, Masato},
     title = {The space of nonextendable quasimorphisms},
     journal = {Algebraic and Geometric Topology},
     pages = {1169--1226},
     year = {2025},
     volume = {25},
     number = {2},
     doi = {10.2140/agt.2025.25.1169},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2025.25.1169/}
}
TY  - JOUR
AU  - Kawasaki, Morimichi
AU  - Kimura, Mitsuaki
AU  - Maruyama, Shuhei
AU  - Matsushita, Takahiro
AU  - Mimura, Masato
TI  - The space of nonextendable quasimorphisms
JO  - Algebraic and Geometric Topology
PY  - 2025
SP  - 1169
EP  - 1226
VL  - 25
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2025.25.1169/
DO  - 10.2140/agt.2025.25.1169
ID  - 10_2140_agt_2025_25_1169
ER  - 
%0 Journal Article
%A Kawasaki, Morimichi
%A Kimura, Mitsuaki
%A Maruyama, Shuhei
%A Matsushita, Takahiro
%A Mimura, Masato
%T The space of nonextendable quasimorphisms
%J Algebraic and Geometric Topology
%D 2025
%P 1169-1226
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2025.25.1169/
%R 10.2140/agt.2025.25.1169
%F 10_2140_agt_2025_25_1169
Kawasaki, Morimichi; Kimura, Mitsuaki; Maruyama, Shuhei; Matsushita, Takahiro; Mimura, Masato. The space of nonextendable quasimorphisms. Algebraic and Geometric Topology, Tome 25 (2025) no. 2, pp. 1169-1226. doi: 10.2140/agt.2025.25.1169

[1] T Akita, Y Liu, Second mod 2 homology of Artin groups, Algebr. Geom. Topol. 18 (2018) 547 | DOI

[2] U Bader, A Lubotzky, R Sauer, S Weinberger, Stability and instability of lattices in semisimple groups, J. Anal. Math. 151 (2023) 1 | DOI

[3] U Bader, R Sauer, Higher Kazhdan property and unitary cohomology of arithmetic groups, preprint (2023)

[4] A Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978) 174 | DOI

[5] A Banyaga, The structure of classical diffeomorphism groups, 400, Kluwer (1997) | DOI

[6] C Bavard, Longueur stable des commutateurs, Enseign. Math. 37 (1991) 109

[7] B Bekka, P De La Harpe, Unitary representations of groups, duals, and characters, 250, Amer. Math. Soc. (2020)

[8] B Bekka, P De La Harpe, A Valette, Kazhdan’s property (T), 11, Cambridge Univ. Press (2008) | DOI

[9] M Bestvina, K Bromberg, K Fujiwara, Bounded cohomology with coefficients in uniformly convex Banach spaces, Comment. Math. Helv. 91 (2016) 203 | DOI

[10] M Bestvina, K Bromberg, K Fujiwara, Stable commutator length on mapping class groups, Ann. Inst. Fourier (Grenoble) 66 (2016) 871 | DOI

[11] M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85

[12] M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69 | DOI

[13] R Bieri, Mayer–Vietoris sequences for HNN-groups and homological duality, Math. Z. 143 (1975) 123 | DOI

[14] A Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1974) 235 | DOI

[15] A Borel, Stable and L2-cohomology of arithmetic groups, Bull. Amer. Math. Soc. 3 (1980) 1025 | DOI

[16] A Borel, Stable real cohomology of arithmetic groups, II, from: "Manifolds and Lie groups", Progr. Math. 14, Birkhäuser (1981) 21 | DOI

[17] J Bowden, S W Hensel, R Webb, Quasi-morphisms on surface diffeomorphism groups, J. Amer. Math. Soc. 35 (2022) 211 | DOI

[18] M Brandenbursky, J Kędra, Fragmentation norm and relative quasimorphisms, Proc. Amer. Math. Soc. 150 (2022) 4519 | DOI

[19] K S Brown, Cohomology of groups, 87, Springer (1982) | DOI

[20] M Bucher, N Monod, The bounded cohomology of SL2 over local fields and S-integers, Int. Math. Res. Not. 2019 (2019) 1601 | DOI

[21] D Burago, S Ivanov, L Polterovich, Conjugation-invariant norms on groups of geometric origin, from: "Groups of diffeomorphisms", Adv. Stud. Pure Math. 52, Math. Soc. Japan (2008) 221 | DOI

[22] M Burger, N Monod, Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. 1 (1999) 199 | DOI

[23] M Burger, N Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002) 219 | DOI

[24] M Burger, S Mozes, Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 747 | DOI

[25] D Calegari, Circular groups, planar groups, and the Euler class, from: "Proceedings of the Casson Fest", Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 431 | DOI

[26] D Calegari, scl, 20, Math. Soc. Japan (2009) | DOI

[27] D Calegari, N Monden, M Sato, On stable commutator length in hyperelliptic mapping class groups, Pacific J. Math. 272 (2014) 323 | DOI

[28] P Capovilla, C Löh, M Moraschini, Amenable category and complexity, Algebr. Geom. Topol. 22 (2022) 1417 | DOI

[29] T D Cochran, S Harvey, P D Horn, Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders, Int. Math. Res. Not. 2012 (2012) 3311 | DOI

[30] M De Chiffre, L Glebsky, A Lubotzky, A Thom, Stability, cohomology vanishing, and nonapproximable groups, Forum Math. Sigma 8 (2020) | DOI

[31] K Dekimpe, M Hartl, S Wauters, A seven-term exact sequence for the cohomology of a group extension, J. Algebra 369 (2012) 70 | DOI

[32] H Endo, D Kotschick, Bounded cohomology and non-uniform perfection of mapping class groups, Invent. Math. 144 (2001) 169 | DOI

[33] M Entov, L Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv. 81 (2006) 75 | DOI

[34] D B A Epstein, K Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997) 1275 | DOI

[35] A Fathi, Structure of the group of homeomorphisms preserving a good measure on a compact manifold, Ann. Sci. École Norm. Sup. 13 (1980) 45 | DOI

[36] F Fournier-Facio, Y Lodha, Second bounded cohomology of groups acting on 1-manifolds and applications to spectrum problems, Adv. Math. 428 (2023) 109162 | DOI

[37] F Fournier-Facio, C Löh, M Moraschini, Bounded cohomology and binate groups, J. Aust. Math. Soc. 115 (2023) 204 | DOI

[38] F Fournier-Facio, C Löh, M Moraschini, Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability, Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (2024) 1169 | DOI

[39] F Fournier-Facio, R D Wade, Aut-invariant quasimorphisms on groups, Trans. Amer. Math. Soc. 376 (2023) 7307 | DOI

[40] R Frigerio, Bounded cohomology of discrete groups, 227, Amer. Math. Soc. (2017) | DOI

[41] R Frigerio, M B Pozzetti, A Sisto, Extending higher-dimensional quasi-cocycles, J. Topol. 8 (2015) 1123 | DOI

[42] K Fujiwara, T Soma, Bounded classes in the cohomology of manifolds, Geom. Dedicata 92 (2002) 73 | DOI

[43] S M Gersten, A presentation for the special automorphism group of a free group, J. Pure Appl. Algebra 33 (1984) 269 | DOI

[44] L Glebsky, A Lubotzky, N Monod, B Rangarajan, Asymptotic cohomology and uniform stability for lattices in semisimple groups, preprint (2023)

[45] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982) 5

[46] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 | DOI

[47] R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597 | DOI

[48] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[49] M Hull, D Osin, Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol. 13 (2013) 2635 | DOI

[50] T Ishida, Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk, PhD thesis, University of Tokyo (2013)

[51] T Ishida, Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups, Proc. Amer. Math. Soc. Ser. B 1 (2014) 43 | DOI

[52] T Ishida, Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups, from: "Geometry, dynamics, and foliations", Adv. Stud. Pure Math. 72, Math. Soc. Japan (2017) 387 | DOI

[53] N V Ivanov, Leray theorems in bounded cohomology theory, preprint (2020)

[54] M Kaluba, D Kielak, P W Nowak, On property (T) for Aut(Fn) and SLn(Z), Ann. of Math. 193 (2021) 539 | DOI

[55] M Kaluba, P W Nowak, N Ozawa, Aut(F5) has property (T), Math. Ann. 375 (2019) 1169 | DOI

[56] M Kawasaki, Relative quasimorphisms and stably unbounded norms on the group of symplectomorphisms of the Euclidean spaces, J. Symplectic Geom. 14 (2016) 297 | DOI

[57] M Kawasaki, Bavard’s duality theorem on conjugation-invariant norms, Pacific J. Math. 288 (2017) 157 | DOI

[58] M Kawasaki, M Kimura, Ĝ-invariant quasimorphisms and symplectic geometry of surfaces, Israel J. Math. 247 (2022) 845 | DOI

[59] M Kawasaki, M Kimura, S Maruyama, T Matsushita, M Mimura, Coarse group theoretic study on stable mixed commutator length, preprint (2023)

[60] M Kawasaki, M Kimura, T Matsushita, M Mimura, Bavard’s duality theorem for mixed commutator length, Enseign. Math. 68 (2022) 441 | DOI

[61] M Kawasaki, M Kimura, T Matsushita, M Mimura, Commuting symplectomorphisms on a surface and the flux homomorphism, Geom. Funct. Anal. 33 (2023) 1322 | DOI

[62] M Kawasaki, S Maruyama, On boundedness of characteristic class via quasi-morphism, J. Topol. Anal. (2024) | DOI

[63] J Kędra, D Kotschick, S Morita, Crossed flux homomorphisms and vanishing theorems for flux groups, Geom. Funct. Anal. 16 (2006) 1246 | DOI

[64] M Kimura, Conjugation-invariant norms on the commutator subgroup of the infinite braid group, J. Topol. Anal. 10 (2018) 471 | DOI

[65] D Kotschick, S Morita, Characteristic classes of foliated surface bundles with area-preserving holonomy, J. Differential Geom. 75 (2007) 273

[66] K Li, Bounded cohomology of classifying spaces for families of subgroups, Algebr. Geom. Topol. 23 (2023) 933 | DOI

[67] A V Malyutin, Operators in the spaces of pseudocharacters of braid groups, Algebra i Analiz 21 (2009) 136

[68] K Mann, Unboundedness of some higher Euler classes, Algebr. Geom. Topol. 20 (2020) 1221 | DOI

[69] S Maruyama, T Matsushita, M Mimura, Invariant quasimorphisms for groups acting on the circle and non-equivalence of SCL, (2022)

[70] S Matsumoto, S Morita, Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc. 94 (1985) 539 | DOI

[71] J Mccool, A faithful polynomial representation of OutF3, Math. Proc. Cambridge Philos. Soc. 106 (1989) 207 | DOI

[72] J Milnor, Introduction to algebraic K-theory, 72, Princeton Univ. Press (1971) | DOI

[73] I Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal. 11 (2001) 807 | DOI

[74] N Monod, Continuous bounded cohomology of locally compact groups, 1758, Springer (2001) | DOI

[75] N Monod, Stabilization for SLn in bounded cohomology, from: "Discrete geometric analysis", Contemp. Math. 347, Amer. Math. Soc. (2004) 191 | DOI

[76] N Monod, Vanishing up to the rank in bounded cohomology, Math. Res. Lett. 14 (2007) 681 | DOI

[77] N Monod, On the bounded cohomology of semi-simple groups, S-arithmetic groups and products, J. Reine Angew. Math. 640 (2010) 167 | DOI

[78] N Monod, Lamplighters and the bounded cohomology of Thompson’s group, Geom. Funct. Anal. 32 (2022) 662 | DOI

[79] N Monod, S Nariman, Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups, Invent. Math. 232 (2023) 1439 | DOI

[80] N Monod, Y Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, J. Differential Geom. 67 (2004) 395

[81] A Monzner, N Vichery, F Zapolsky, Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization, J. Mod. Dyn. 6 (2012) 205 | DOI

[82] M Moraschini, G Raptis, Amenability and acyclicity in bounded cohomology, Rev. Mat. Iberoam. 39 (2023) 2371 | DOI

[83] J Neukirch, A Schmidt, K Wingberg, Cohomology of number fields, 323, Springer (2008) | DOI

[84] W D Neumann, L Reeves, Central extensions of word hyperbolic groups, Ann. of Math. 145 (1997) 183 | DOI

[85] B B Newman, Some results on one-relator groups, Bull. Amer. Math. Soc. 74 (1968) 568 | DOI

[86] M Nitsche, Computer proofs for property (T), and SDP duality, preprint (2020)

[87] L Polterovich, The geometry of the group of symplectic diffeomorphisms, Birkhäuser (2001) | DOI

[88] L Polterovich, D Rosen, Function theory on symplectic manifolds, 34, Amer. Math. Soc. (2014) | DOI

[89] P Py, Quasi-morphismes et invariant de Calabi, Ann. Sci. École Norm. Sup. 39 (2006) 177 | DOI

[90] A I Shtern, Extension of pseudocharacters from normal subgroups, III, Proc. Jangjeon Math. Soc. 19 (2016) 609

[91] W Thurston, Noncobordant foliations of S3, Bull. Amer. Math. Soc. 78 (1972) 511 | DOI

[92] W P Thurston, On the structure of the group of volume preserving diffeomorphisms, preprint (1973)

[93] W P Thurston, Hyperbolic structures on 3-manifolds, II : Surface groups and 3-manifolds which fiber over the circle, preprint (1986)

[94] B Tshishiku, Borel’s stable range for the cohomology of arithmetic groups, J. Lie Theory 29 (2019) 1093

[95] T Tsuboi, On the uniform perfectness of diffeomorphism groups, from: "Groups of diffeomorphisms", Adv. Stud. Pure Math. 52, Math. Soc. Japan (2008) 505 | DOI

[96] T Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Comment. Math. Helv. 87 (2012) 141 | DOI

[97] T Tsuboi, Several problems on groups of diffeomorphisms, from: "Geometry, dynamics, and foliations", Adv. Stud. Pure Math. 72, Math. Soc. Japan (2017) 239 | DOI

Cité par Sources :