Geometry and Topology Seminar: A notion of extended Theta-positivity representation

We study a notion of extended positive representations of surface groups. Examples include maximal representations (Burger—Iozzi—Wienhard), positive representations (Fock—Goncharov), and cusped Hitchin representations (Canary—Zhang—Zimmer). We discuss conditions under which these representations are Anosov or relatively Anosov. We prove that extended positivity is a closed condition, and open in certain subspace of the character variety. Moreover, we describe the boundary of the closure of extended positive representations into a semisimple Lie group G in the real spectral compactification of Hom(\Gamma,G) (introduced by Burger—Iozzi—Parreau—Pozzetti), showing that it consists of extended positive representations into extensions of G over real closed fields. This is joint work with Xenia Flamm, Nicolas Tholozan, and Tengren Zhang.