Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

Abstract

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d, R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu. Contents 1. Introduction 1 2. Preliminaries 4 3. Anosov representations into SL(d, K) 6 4. Basic properties of Anosov representations 9 5. Basic properties of cusp representations 13 6. A dynamical characterization of linear Anosov representations 15 7. Hitchin representations are Borel Anosov 20 8. Stability of Anosov representations 22 9. Positive representations in the sense of Fock-Goncharov 29 Appendix A. Anosov representations into semisimple Lie groups 33 References 38

Entropy rigidity for cusped Hitchin representations

“…Cusped Anosov representations of geometrically finite Fuchsian groups. Cusped Anosov representations of geometrically finite Fuchsian groups were introduced in [19] as natural generalizations of Anosov representations which take parabolic elements to elements whose (generalized) eigenvalues all have modulus 1. These representations are also relatively Anosov in the sense of Kapovich-Leeb [32] and relatively dominated in the sense of Zhu [56].…”

“…(To be precise, in [19] Anosov representations were defined in terms of the exponential contraction of a linear flow on a vector bundle associated to a representation and this was shown to be equivalent to the definition above.) If Γ contains a parabolic element, we refer to such representations as cusped Anosov when we want to distinguish them from traditional Anosov representations (which cannot contain unipotent elements in their image).…”

“…For any x, y ∈ H 2 , let d(x, y) denote the hyperbolic distance between x and y, and for any γ ∈ PSL(2, R), let (γ) denote the minimal translation distance of γ acting on H 2 . Theorem 2.2 (Canary-Zhang-Zimmer [19]). If Γ is a geometrically finite Fuchsian group, ρ : Γ → PGL(d, K) is P θ -Anosov and b 0 ∈ H 2 , then…”

“…into the space F of complete d-dimensional flags. One may then naturally define a representation ρ of a Fuchsian group Γ ⊂ PSL(2, R) into PSL(d, R) to be Hitchin if there is a continuous positive ρ-equivariant map of the limit set of Γ into F. Hitchin representations of geometrically finite Fuchsian groups share many of the same properties as classical Hitchin representations (see Canary-Zhang-Zimmer [19]).…”

“…( In the process of establishing our main result, we introduce the class of transverse subgroups of PGL(d, R) which includes all Anosov subgroups, all images of Hitchin representations, all images of cusped Anosov representations of geometrically finite Fuchsian groups in the sense of [19], all images of relatively dominated representations in the sense of Zhu [56], all images of relatively Anosov representations in the sense of Kapovich-Leeb [32], all subgroups of these groups, all discrete subgroups of PO(d − 1, 1), and all discrete groups of projective automorphisms that preserve strictly convex domains with C 1 boundary in real projective space. We note that the definition of transverse groups and many of general results we establish about them do not assume finite generation.…”