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Positivity and representations of surface groups
- Part of
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- Journal:
- Forum of Mathematics, Pi / Volume 14 / 2026
- Published online by Cambridge University Press:
- 09 February 2026, e6
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- Article
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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.
