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.Résumé. -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. ANNALES DE L'INSTITUT FOURIERRELATIVELY DOMINATED REPRESENTATIONS 5 Relatively hyperbolic groupsRelative hyperbolicity is a group-theoretic notion-originally suggested by Gromov in [14], and further developed by Bowditch [6], Farb [11], Yaman [25], Groves-Manning [15], and others-of non-positive curvature inspired by the geometry of cusped hyperbolic manifolds and free products.The geometry of a relatively hyperbolic group is akin to the geometry of a cusped hyperbolic manifold in that it is negatively-curved outside of certain regions, which, like the cusps in a cusped hyperbolic manifold, can be more or less separated from each other.There are various ways to make this intuition precise, resulting in various equivalent characterizations of relatively hyperbolic groups. We will use a definition of Bowditch, in the tradition of Gromov:Consider a finite-volume cusped hyperbolic manifold with an open neighborhood of each cusp removed: call the resulting truncated manifold M . The universal cover M of such a M is hyperbolic space with a countable set of horoballs removed. The universal cover M is not Gromov-hyperbolic; distances along horospheres that bound removed horoballs are distorted. If we glue the removed horoballs back in to the universal cover, however, the resulting space will again be hyperbolic space.We can do a similar thing from a gr...
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.
Relatively dominated representations give a common generalization of geometrically finiteness in rank one on the one hand, and the Anosov condition which serves as a higher-rank analogue of convex cocompactness on the other. This note proves three results about these representations.Firstly, we remove the quadratic gaps assumption involved in the original definition. Secondly, we give a characterization using eigenvalue gaps, providing a relative analogue of a result of Kassel-Potrie for Anosov representations. Thirdly, we formulate characterizations in terms of singular value or eigenvalue gaps combined with limit maps, in the spirit of Guéritaud-Guichard-Kassel-Wienhard for Anosov representations, and use them to show that inclusion representations of certain groups playing weak ping-pong and positive representations in the sense of Fock-Goncharov are relatively dominated.
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