Positivity and representations of surface groups

Abstract In [24, 26] Guichard and Wienhard introduced the notion of $\Theta $ -positivity, a generalization of Lusztig’s total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $\Theta $ -positive representations of surface groups. We prove that $\Theta $ -positive representations of closed surface groups are $\Theta $ -Anosov. This implies that $\Theta $ -positive representations are discrete and faithful and that the set of $\Theta $ -positive representations is open in the representation variety. We further establish important properties on limits of $\Theta $ -positive representations, proving that the set of $\Theta $ -positive representations is closed in the set of representations containing a $\Theta $ -proximal element. This is used in [3] to prove the closedness of the set of $\Theta $ -positive representations.