Relatively Anosov representations via flows. II: Examples (Q6561013)

The paper is the second in a series of two papers that develops a theory of relatively Anosov representations for relatively hyperbolic groups using the original ``contracting flow on a bundle'' definition of Anosov representations introduced by \textit{F. Labourie} [Invent. Math. 165, No. 1, 51--114 (2006; Zbl 1103.32007)] and \textit{O. Guichard} and \textit{A. Wienhard} [Invent. Math. 190, No. 2, 357--438 (2012; Zbl 1270.20049)]. For Part I of this series see [the authors, ``Relatively Anosov representations via flows. I: Theory'', Preprint, \url{arXiv:2207.14737}]. In this paper, the authors focus on building examples.\N\NThe first examples support the idea that relatively Anosov representations are a higher rank generalization of geometrically finite representations into rank-one semisimple Lie groups. Let \(G\) be the connected component of the identity in the isometry group of a negatively curved symmetric space \(X\). Let \(\Gamma \leq G\) be a geometrically finite group and denote by \(\mathcal{C}_X(\Gamma)\) the convex hull of its limit set in \(X\). The first result states that restricting a representation \(\tau : G \to \mathrm{SL}(d,\mathbb{K})\) (for \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\)) whose image contains a \(P_k\)-proximal element to \(\Gamma\), yields a representation of \(\Gamma\) that is uniformly \(P_k\)-Anosov relative to \(\mathcal{C}_X(\Gamma)\). Furthermore, using the stability property of relatively Anosov representations established in Part I of the series [the authors, loc. cit.], a small deformation of this representation is still relatively Anosov.\N\NThe authors now apply their general results in the context of convex projective geometry. More precisely they characterize when the image of a relatively \(P_1\)-Anosov representation into \(\mathrm{PGL}(d,\mathbb{R})\) is a \emph{projectively visible subgroup} of the automorphism group of some properly convex domain, that acts geometrically finitely on its limit set. This characterization is in terms of a \emph{lifting property} of the representation to \(\mathrm{SL}^\pm(d,\mathbb{R})\). As in the general case, the authors produce examples of such representations from geometrically finite subgroups \(\Gamma \leq G\) using the restriction of \(P_1\)-proximal representations from \(G\) to \(\mathrm{PGL}(d,\mathbb{R})\).\N\NThe final results concern a collection of other examples. The authors characterize (up to finite index) which linear discrete groups appear as the image of a peripheral subgroup under a relatively \(P_k\)-Anosov representation. They furthermore show that certain representations of \(\mathrm{PSL}(2,\mathbb{Z})\) in \(\mathrm{PGL}(3,\mathbb{R})\) defined by \textit{R. Schwartz} [Publ. Math., Inst. Hautes Étud. Sci. 78, 187--206 (1993; Zbl 0806.57004)] are relatively \(P_1\)-Anosov. The paper ends with an example of a representation of a relatively hyperbolic group whose semisimplification is relatively Anosov, but which itself is not relatively Anosov.