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…

• NS
  National Science Foundation

  Award: DMS 1107452, 1107263, 1107367, 1440140
• DF
  Deutsche Forschungsgemeinschaft

  Award: EXC-2181/1 - 390900948
• ER
  European Research Council

  Award: 614733, 101018839, 101095722