Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms
[Structures équivariantes hiérarchiquement hyperboliques pour les groupes de 3-variétés via les quasimorphismes]
Hagen, Mark1 ; Russell, Jacob2 ; Sisto, Alessandro3 ; Spriano, Davide4
1 School of Mathematics, University of Bristol, Bristol (UK)
2 Department of Mathematics, Rice University, Houston, TX (USA)
3 Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh (UK)
4 Mathematical Institute, University of Oxford, Oxford (UK)
Annales de l'Institut Fourier, Online first, 60 p.

Behrstock, Hagen et Sisto ont classifié les groupes fondamentaux des 3-variétés qui admettent une structure d’espace hyperbolique hiérarchique. Mais ces structures e sont pas toujours équivariantes. Dans cet article, nous classifions les groupes fondamentaux des 3-variétés admettant des structures HHS équivariantes. L’élément clé de notre preuve est que les groupes admissibles introduits par Croke et Kleiner admettent toujours des structures HHS équivariantes. Pour les variétés de graphes non géométriques, cela est contraire à une conjecture de Behrstock, Hagen et Sisto et contraste également avec des résultats sur les structures cubiques sur ces groupes. Nos arguments impliquent la construction de quasimorphismes appropriés sur les pièces de Seifert, afin de construire des actions sur des quasi-lignes.

Behrstock, Hagen and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups admitting equivariant hierarchically hyperbolic structures. The key component of our proof is that the admissible groups introduced by Croke and Kleiner always admit equivariant hierarchically hyperbolic structures. For non-geometric graph manifolds, this is contrary to a conjecture of Behrstock, Hagen and Sisto and also contrasts with results about CAT(0) cubical structures on these groups. Perhaps surprisingly, our arguments involve the construction of suitable quasimorphisms on the Seifert pieces, in order to construct actions on quasi-lines.

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Révisé le :
Accepté le :
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DOI : 10.5802/aif.3654
Classification : 20F67, 57K35
Keywords: Hierarchically hyperbolic group, 3-manifold.
Mot clés : groupe hiérarchique hyperbolic, 3-variété.
Hagen, Mark 1 ; Russell, Jacob 2 ; Sisto, Alessandro 3 ; Spriano, Davide 4

1 School of Mathematics, University of Bristol, Bristol (UK)
2 Department of Mathematics, Rice University, Houston, TX (USA)
3 Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh (UK)
4 Mathematical Institute, University of Oxford, Oxford (UK) 
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Hagen, Mark; Russell, Jacob; Sisto, Alessandro; Spriano, Davide. Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms. Annales de l'Institut Fourier, Online first, 60 p.

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