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

Published online:
DOI: 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
Mot clés : quasimorphisme, longueur stable de commutateurs, cohomologie bornée, complexe cubique CAT(0), groupe d’Artin à angle droit, propriété médiane
Fernós, Talia 1; Forester, Max 2; Tao, Jing 2

1 Department of Mathematics and Statistics University of North Carolina at Greensboro 317 College Avenue Greensboro, NC 27412 (USA)
2 Department of Mathematics University of Oklahoma Norman, OK 73019 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/aif.3277/

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