Effective quasimorphisms on right-angled Artin groups

Fernós, Talia; Forester, Max; Tao, Jing

Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1575-1626.

Nous construisons des nouvelles familles de quasi-morphismes sur de nombreux groupes agissant sur des complexes cubiques CAT(0). Ces quasi-morphismes ont leur défaut majoré par 12, et sont suffisamment nombreux pour « voir » tous les éléments qui agissent de manière hyperbolique sur le complexe cubique. Nous déduisons que la longueur stable des commutateurs de tous ces éléments est minorée par 1/24.

Les actions pour lesquelles ces résultats sont vérifiés comprennent l’action standard d’un groupe d’Artin à angles droits sur son complexe cubique associé. En particulier, la longueur stable des commutateurs de tout élément non trivial d’un groupe d’Artin à angles droits est minorée par 1/24.

Ces résultats reposent sur de nouveaux outils que nous développons pour étudier les actions de groupes sur des complexes cubiques CAT(0) : l’ensemble caractéristique essentiel et les plongements euclidiens équivariants.

We construct families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they “see” all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24.

The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24.

These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings.

Classification : 20F65, 20F67, 57M07
Keywords: quasimorphism, stable commutator length, bounded cohomology, CAT(0) cube complex, right-angled Artin group, median property
Mot clés : quasimorphisme, longueur stable de commutateurs, cohomologie bornée, complexe cubique CAT(0), groupe d’Artin à angle droit, propriété médiane

[1] Bavard, Christophe Longueur stable des commutateurs, Enseign. Math., Volume 37 (1991) no. 1-2, pp. 109-150 | MR | Zbl

[2] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Bounded cohomology with coefficients in uniformly convex Banach spaces, Comment. Math. Helv., Volume 91 (2016) no. 2, pp. 203-218 | DOI | MR | Zbl

[3] Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji Stable commutator length on mapping class groups, Ann. Inst. Fourier, Volume 66 (2016) no. 3, pp. 871-898 | DOI | MR | Zbl

[4] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | DOI | MR | Zbl

[5] Brodzki, Jacek; Campbell, S. J.; Guentner, Erik; Niblo, Graham; Wright, Nick J. Property A and $\mathrm{CAT}\left(0\right)$ cube complexes, J. Funct. Anal., Volume 256 (2009) no. 5, pp. 1408-1431 | DOI | MR | Zbl

[6] Brooks, Robert Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, 1978) (Annals of Mathematics Studies), Volume 97, Princeton University Press, 1981, pp. 53-63 | DOI | MR | Zbl

[7] Calegari, Danny scl, MSJ Memoirs, 20, Mathematical Society of Japan, 2009, xii+209 pages | DOI | MR | Zbl

[8] Calegari, Danny; Fujiwara, Koji Stable commutator length in word-hyperbolic groups, Groups Geom. Dyn., Volume 4 (2010) no. 1, pp. 59-90 | DOI | MR | Zbl

[9] Caprace, Pierre-Emmanuel; Sageev, Michah Rank rigidity for $\mathrm{CAT}\left(0\right)$ cube complexes, Geom. Funct. Anal., Volume 21 (2011) no. 4, pp. 851-891 | DOI | MR | Zbl

[10] Chatterji, Indira; Fernós, Talia; Iozzi, Alessandra The median class and superrigidity of actions on $\mathrm{CAT}\left(0\right)$ cube complexes, J. Topol., Volume 9 (2016) no. 2, pp. 349-400 (With an appendix by Pierre-Emmanuel Caprace) | DOI | MR | Zbl

[11] Chatterji, Indira; Niblo, Graham From wall spaces to $\mathrm{CAT}\left(0\right)$ cube complexes, Int. J. Algebra Comput., Volume 15 (2005) no. 5-6, pp. 875-885 | DOI | MR | Zbl

[12] Clay, Matt; Forester, Max; Louwsma, Joel Stable commutator length in Baumslag–Solitar groups and quasimorphisms for tree actions, Trans. Am. Math. Soc., Volume 368 (2016) no. 7, pp. 4751-4785 | DOI | MR | Zbl

[13] Davis, Michael W. The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, 32, Princeton University Press, 2008, xvi+584 pages | MR | Zbl

[14] Duncan, Andrew J.; Howie, James The genus problem for one-relator products of locally indicable groups, Math. Z., Volume 208 (1991) no. 2, pp. 225-237 | DOI | MR | Zbl

[15] Epstein, David B. A.; Fujiwara, Koji The second bounded cohomology of word-hyperbolic groups, Topology, Volume 36 (1997) no. 6, pp. 1275-1289 | DOI | MR | Zbl

[16] Fernós, Talia The Furstenberg–Poisson boundary and $\mathrm{CAT}\left(0\right)$ cube complexes, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 6, pp. 2180-2223 | DOI | MR | Zbl

[17] Gromov, Michael Volume and bounded cohomology, Publ. Math., Inst. Hautes Étud. Sci. (1982) no. 56, pp. 5-99 | MR | Zbl

[18] Haglund, Frédéric Isometries of $\mathrm{CAT}\left(0\right)$ cube complexes are semi-simple (2007) (http://arxiv.org/abs/0705.3386)

[19] Haglund, Frédéric; Wise, Daniel T. Special cube complexes, Geom. Funct. Anal., Volume 17 (2008) no. 5, pp. 1551-1620 | DOI | MR | Zbl

[20] Hull, Michael; Osin, Denis Induced quasicocycles on groups with hyperbolically embedded subgroups, Algebr. Geom. Topol., Volume 13 (2013) no. 5, pp. 2635-2665 | DOI | MR | Zbl

[21] Koberda, Thomas Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups, Geom. Funct. Anal., Volume 22 (2012) no. 6, pp. 1541-1590 | DOI | MR | Zbl

[22] Mitsumatsu, Yoshihiko Bounded cohomology and $l^1$-homology of surfaces, Topology, Volume 23 (1984) no. 4, pp. 465-471 | DOI | MR | Zbl

[23] Nica, Bogdan Cubulating spaces with walls, Algebr. Geom. Topol., Volume 4 (2004), pp. 297-309 | DOI | MR | Zbl

[24] Roller, Martin Poc sets, median algebras and group actions. An extended study of Dunwoody’s construction and Sageev’s theorem (1998) (Habilitationschrift, Regensberg)

[25] Sageev, Michah Ends of group pairs and non-positively curved cube complexes, Proc. Lond. Math. Soc., Volume 71 (1995) no. 3, pp. 585-617 | DOI | MR | Zbl

[26] Wise, Daniel T. From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, 117, American Mathematical Society, 2012, xiv+141 pages | MR | Zbl