An Introduction to Higher Rank Teichmüller Theory - 2/4

Explore discrete subgroups of semisimple Lie groups through this mathematical lecture that forms the second part of a four-part series on higher rank Teichmüller theory. Delve into the deformation theory of discrete subgroups isomorphic to fundamental groups of surfaces, examining how these can be parametrized as subsets of the character variety X=Hom(Γ, G)/G. Learn about the Anosov condition and its role in describing open subsets of the character variety, then advance to higher rank Teichmüller theories focusing on connected components consisting entirely of discrete and faithful representations. Discover the Θ-positivity framework developed by Guichard-Wienhard for classical groups G, which provides a Lie algebraic explanation for these phenomena. Understand how closedness in the character variety connects to a generalized collar lemma that extends key geometric features from hyperbolic surfaces to higher rank settings. The presentation builds on collaborative research with Beyrer, Guichard, Labourie, and Wienhard, offering insights into the intersection of geometric group theory, Lie theory, and surface topology.