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We study a notion of extended positive representations of general Fuchsian groups. Examples include Fock-Goncharov’s positive representations, maximal representations, Theta-positive representations, and cusped Hitchin representations. We discuss conditions under which these representations are Anosov, relatively Anosov, or extended geometrically finite. We prove that extended positivity is a closed condition, and open in a certain subspace of the character variety. Moreover, we describe the boundary of the closure of extended positive representations into a semisimple Lie group in the real spectral compactification of the representation variety, showing that it consists of extended positive representations into extensions of the Lie group over real closed fields. In particular, we show that the set of positive representations is semi-algebraic. This is joint work with Xenia Flamm, Nicolas Tholozan, and Tengren Zhang.

Extended positive representations over real closed fields