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Relatively dominated representations from eigenvalue gaps and limit maps

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 technical assumption of quadratic gaps 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.