𝐴𝑢𝑡-invariant quasimorphisms on groups

Fournier-Facio, Francesco; Wade, Richard

doi:10.1090/tran/8980

$\operatorname {Aut}$-invariant quasimorphisms on groups
HTML articles powered by AMS MathViewer

by Francesco Fournier-Facio and Richard D. Wade;

Trans. Amer. Math. Soc. 376 (2023), 7307-7327

DOI: https://doi.org/10.1090/tran/8980

Published electronically: June 21, 2023

HTML | PDF | Request permission

Abstract:

For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian.

This was known for $F_2$ by a result of Brandenbursky and Marcinkowski [Comment. Math. Helv. 94 (2019), pp. 661–687], but is new even for free groups of higher rank, settling a question of Miklós Abért. The case of graph products of finitely generated abelian groups settles a question of Michał Marcinkowski. As a consequence, we deduce that a variety of $Aut$-invariant norms on such groups are unbounded.

References

M. Abért, Some questions, https://www.renyi.hu/~abert/questions.pdf, 2010.
Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, Stable commutator length on mapping class groups, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 871–898 (English, with English and French summaries). MR 3494163, DOI 10.5802/aif.3028
M. Bestvina, R. Dickmann, G. Domat, S. Kwak, P. Patel, and E. Stark. Free products from spinning and rotating families. arXiv:2010.10735, to appear in Enseign. Math., 2020.
Mladen Bestvina and Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89. MR 1914565, DOI 10.2140/gt.2002.6.69
Mladen Bestvina and Koji Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (2009), no. 1, 11–40. MR 2507218, DOI 10.1007/s00039-009-0717-8
Mladen Bestvina and Mark Feighn, A hyperbolic $\textrm {Out}(F_n)$-complex, Groups Geom. Dyn. 4 (2010), no. 1, 31–58. MR 2566300, DOI 10.4171/GGD/74
B. Brück, F. Fournier-Facio, and C. Löh, Median quasimorphisms on CAT(0) cube complexes and their cup products, Preprint, arXiv:2209.05811, 2022.
Michael Brandenbursky and MichałMarcinkowski, Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups, Comment. Math. Helv. 94 (2019), no. 4, 661–687. MR 4046001, DOI 10.4171/cmh/470
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., No. 97, Princeton Univ. Press, Princeton, NJ, 1981, pp. 53–63. MR 624804
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Valery Bardakov, Vladimir Shpilrain, and Vladimir Tolstykh, On the palindromic and primitive widths of a free group, J. Algebra 285 (2005), no. 2, 574–585. MR 2125453, DOI 10.1016/j.jalgebra.2004.11.003
Danny Calegari, What is$\ldots$ stable commutator length?, Notices Amer. Math. Soc. 55 (2008), no. 9, 1100–1101. MR 2451345
Danny Calegari, scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, 2009. MR 2527432, DOI 10.1142/e018
Pierre-Emmanuel Caprace and Koji Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), no. 5, 1296–1319. MR 2585575, DOI 10.1007/s00039-009-0042-2
I. Chifan, A. Ioana, D. Osin, and B. Sun, Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces, Preprint, arXiv:2304.07455, 2023.
F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI 10.1090/memo/1156
David B. A. Epstein and Koji Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), no. 6, 1275–1289. MR 1452851, DOI 10.1016/S0040-9383(96)00046-8
Talia Fernós, Max Forester, and Jing Tao, Effective quasimorphisms on right-angled Artin groups, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 4, 1575–1626 (English, with English and French summaries). MR 4010865, DOI 10.5802/aif.3277
D. I. Fouxe-Rabinovitch, Über die Automorphismengruppen der freien Produkte. I, Rec. Math. [Mat. Sbornik] N.S. 8(50) (1940), 265–276 (Russian, with German summary). MR 3413
D. I. Fouxe-Rabinovitch, Über die Automorphismengruppen der freien Produkte. II, Rec. Math. [Mat. Sbornik] N.S. 9(51) (1941), 183–220 (Russian, with German summary). MR 4625
Roberto Frigerio, Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, 2017. MR 3726870, DOI 10.1090/surv/227
Koji Fujiwara, The second bounded cohomology of a group acting on a Gromov-hyperbolic space, Proc. London Math. Soc. (3) 76 (1998), no. 1, 70–94. MR 1476898, DOI 10.1112/S0024611598000033
Koji Fujiwara, The second bounded cohomology of an amalgamated free product of groups, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1113–1129. MR 1491864, DOI 10.1090/S0002-9947-99-02282-5
A. Genevois. Automorphisms of graph products of groups and acylindrical hyperbolicity. arXiv:1807.00622, to appear in Mem. Amer. Math. Soc., 2018.
Anthony Genevois, Negative curvature in automorphism groups of one-ended hyperbolic groups, J. Comb. Algebra 3 (2019), no. 3, 305–329. MR 4011668, DOI 10.4171/JCA/33
Anthony Genevois and Camille Horbez, Acylindrical hyperbolicity of automorphism groups of infinitely ended groups, J. Topol. 14 (2021), no. 3, 963–991. MR 4503954, DOI 10.1112/topo.12203
Étienne Ghys, Groupes d’homéomorphismes du cercle et cohomologie bornée, The Lefschetz centennial conference, Part III (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1987, pp. 81–106 (French, with English summary). MR 893858, DOI 10.24033/msmf.358
Vincent Guirardel and Gilbert Levitt, Splittings and automorphisms of relatively hyperbolic groups, Groups Geom. Dyn. 9 (2015), no. 2, 599–663. MR 3356977, DOI 10.4171/GGD/322
A. Genevois and A. Martin, Automorphism groups of cyclic products of groups, Preprint, arXiv:1803.07536, 2018.
R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and geometric group theory (Edinburgh, 1993) London Math. Soc. Lecture Note Ser., vol. 204, Cambridge Univ. Press, Cambridge, 1995, pp. 111–163. MR 1320279
A. Hase, Dynamics of ${O}ut({F}_n)$ on the second bounded cohomology of ${F}_n$, Preprint, arXiv:1805.00366, 2018.
Peter Hegarty, The absolute centre of a group, J. Algebra 169 (1994), no. 3, 929–935. MR 1302126, DOI 10.1006/jabr.1994.1318
Peter V. Hegarty, Autocommutator subgroups of finite groups, J. Algebra 190 (1997), no. 2, 556–562. MR 1441963, DOI 10.1006/jabr.1996.6921
Nicolaus Heuer, Stable commutator length, Bounded cohomology and simplicial volume, London Math. Soc. Lecture Note Ser., vol. 479, Cambridge Univ. Press, Cambridge, 2023, pp. 65–76. MR 4496346
Michael Handel and Lee Mosher, Second bounded cohomology and WWPD, Kyoto J. Math. 61 (2021), no. 4, 873–904. MR 4415399, DOI 10.1215/21562261-2021-0017
Tobias Hartnick and Pascal Schweitzer, On quasioutomorphism groups of free groups and their transitivity properties, J. Algebra 450 (2016), 242–281. MR 3449692, DOI 10.1016/j.jalgebra.2015.10.010
B. Karlhofer, Aut-invariant quasimorphisms on free products, Ann. Math. Qué. 2021, pages 1–19.
B. Karlhofer, Aut-invariant quasimorphisms on graph products of abelian groups, Preprint, arXiv:2107.12171, 2021.
Morimichi Kawasaki and Mitsuaki Kimura, $\hat G$-invariant quasimorphisms and symplectic geometry of surfaces, Israel J. Math. 247 (2022), no. 2, 845–871. MR 4425357, DOI 10.1007/s11856-021-2283-1
M. Kawasaki, M. Kimura, S. Maruyama, T. Matsushita, and M. Mimura, Mixed commutator lengths, wreath products and general ranks, Preprint, arXiv:2203.04048, 2022. Kodai Math J., to appear.
M. Kawasaki, M. Kimura, S. Maruyama, T. Matsushita, and M. Mimura, Survey on invariant quasimorphisms and stable mixed commutator length, Preprint, arXiv:2212.11180, 2022.
Morimichi Kawasaki, Mitsuaki Kimura, Takahiro Matsushita, and Masato Mimura, Bavard’s duality theorem for mixed commutator length, Enseign. Math. 68 (2022), no. 3-4, 441–481. MR 4452430, DOI 10.4171/lem/1037
A. V. Malyutin, Writhe of (closed) braids, Algebra i Analiz 16 (2004), no. 5, 59–91 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 5, 791–813. MR 2106667, DOI 10.1090/S1061-0022-05-00879-4
MichałMarcinkowski, Aut-invariant word norm on right-angled Artin and right-angled Coxeter groups, Michigan Math. J. 69 (2020), no. 2, 285–295. MR 4104374, DOI 10.1307/mmj/1573873440
G. A. Miller, Automorphism commutators, Proc. Natl. Acad. Sci. USA 15 (1929), no. 8, 672–675.
S. Maruyama, T. Matsushita, and M. Mimura, SCL and mixed SCL are not equivalent for surface groups, Preprint, arXiv:2203.09221, 2022.
Ashot Minasyan and Denis Osin, Correction to: Acylindrical hyperbolicity of groups acting on trees [ MR3368093], Math. Ann. 373 (2019), no. 1-2, 895–900. MR 3968890, DOI 10.1007/s00208-018-1699-3
Nicolas Monod, Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, vol. 1758, Springer-Verlag, Berlin, 2001. MR 1840942, DOI 10.1007/b80626
D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
Leonid Polterovich and Daniel Rosen, Function theory on symplectic manifolds, CRM Monograph Series, vol. 34, American Mathematical Society, Providence, RI, 2014. MR 3241729, DOI 10.1090/crmm/034