Effective quasimorphisms on right-angled Artin groups
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1575-1626

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.

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.

Received : 2017-03-02
Revised : 2017-12-04
Accepted : 2018-06-12
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3277
Classification:  20F65,  20F67,  57M07
Keywords: quasimorphism, stable commutator length, bounded cohomology, CAT(0) cube complex, right-angled Artin group, median property
@article{AIF_2019__69_4_1575_0,
     author = {Fern\'os, Talia and Forester, Max and Tao, Jing},
     title = {Effective quasimorphisms on right-angled Artin groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1575-1626},
     doi = {10.5802/aif.3277},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1575_0}
}
Fernós, Talia; Forester, Max; Tao, Jing. Effective quasimorphisms on right-angled Artin groups. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1575-1626. doi : 10.5802/aif.3277. https://aif.centre-mersenne.org/item/AIF_2019__69_4_1575_0/

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