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Explore discrete subgroups of semisimple Lie groups through this third lecture in a four-part minicourse on higher rank Teichmüller theory. Delve into fundamental groups of surfaces and their rich deformation theory as parametrized subsets of character varieties. Learn about the Anosov condition and its role in describing open subsets of character varieties, then examine higher rank Teichmüller theories as connected components consisting solely of discrete and faithful representations. Discover how Θ-positivity, a Lie algebraic framework developed by Guichard-Wienhard, explains these theories for classical groups through research conducted with Beyrer-Guichard-Labourie-Wienhard. Understand the geometric foundations underlying closedness in character varieties through a generalized collar lemma that extends key features of hyperbolic surfaces to higher rank settings.
