Relatively dominated representations
[Représentations relativements dominées]
Annales de l'Institut Fourier, Online first, 67 p.

Les représentations Anosov fournissent une classe de sous-groupes discrets des groupes de Lie qui généralisent les sous-groupes convexes-cocompacts d’un groupe de Lie de rang un. En rang un, la classe des sous-groupes géométriquement finis est une généralisation de la classe des sous-groupes convexes-cocompacts, qui autorise des défauts isolés d’hyperbolicité. Nous introduisons les représentations relativement dominées comme une relativisation des représentations Anosov, autrement dit un analogue de la finitude géométrique en rang supérieur. Nous montrons qu’un groupe qui admet une représentation relativement dominée est nécessairement relativement hyperbolique et que ces représentations induisent des applications de bord satisfaisant des bonnes propriétés. Nous donnons des exemples et faisons des connexions avec le travail de Kapovich–Leeb sur d’ autres analogues de la finitude géometrique en rang supérieur.

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness. We prove that groups admitting relatively dominated representations must be relatively hyperbolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich–Leeb which also introduces higher-rank analogues of geometric finiteness.

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DOI : https://doi.org/10.5802/aif.3449
Classification : 22E40,  20F67,  37D30,  53C35
Mots clés : Sous-groupes discrets de groupes de Lie, finitude géométrique, décompositions dominées, groupes relativement hyperboliques
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Zhu, Feng. Relatively dominated representations. Annales de l'Institut Fourier, Online first, 67 p.

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