Appears in collection : 2025 IHES Summer School – Discrete Subgroups of Lie Groups : Dynamics, Actions, Rigidity

The minicourse will focus on discrete subgroups of semisimple Lie groups G isomorphic to fundamental groups $\Gamma$ of surfaces. These typically admit a rich deformation theory and can be parametrized as subset of the character variety $X=Hom(\Gamma, G)/G$. I will first discuss the Anosov condition, describing open subsets of $X$ and then discuss higher rank Teichmüller theories: connected components of $X$ only consisting of discrete and faithful representations. We proved with Beyrer-Guichard-Labourie-Wienhard that for classical groups G these are explained by $\Theta$-positivity, a Lie algebraic framework introduced by Guichard-Wienhard. After introducing this concept I will explain how closedness in the character variety is ultimately due to a collar lemma, generalizing a key geometric feature of hyperbolic surfaces.