Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.G.

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the "length" of any closed curve on S in terms of the parameters describing this representation. We then show that this lower bound is good enough to produce large families of sequences in the Hitchin component along which the topological entropy converges to 0.Comment: Some typos from previous version correcte

Let M be a compact surface of negative Euler characteristic and let C(M ) be the deformation space of convex real projective structures on M . For every choice of pants decomposition for M , there is a well known parameterization of C(M ) known as the Goldman parameterization. In this paper, we study how some geometric properties of the real projective structure on M degenerate as we deform it so that the internal parameters of the Goldman parameterization leave every compact set while the boundary invariants remain bounded away from zero and infinity.

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

Properly discontinuous actions of a surface group by affine automorphisms of R d were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in SOpn, n´1q, so that the affine action is by isometries of a flat pseudo-Riemannian metric on R d of signature pn, n´1q. Here, the translational part determines a deformation of the linear part into PSOpn, nq-Hitchin representations and the crucial step is to show that such representations are not Anosov in PSLp2n, Rq with respect to the stabilizer of an n-plane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudo-Riemannian hyperbolic space of signature pn, n´1q by a PSOpn, nq-Hitchin representation fails to be properly discontinuous.