An Introduction to Higher Rank Teichmüller Theory - 4/4

Explore the final lecture in a comprehensive minicourse on higher rank Teichmüller theory, focusing on discrete subgroups of semisimple Lie groups G that are isomorphic to fundamental groups of surfaces. Delve into the rich deformation theory of these groups and their parametrization as subsets of the character variety X=Hom(Γ, G)/G. Examine the Anosov condition and its role in describing open subsets of X, then investigate higher rank Teichmüller theories as connected components of X consisting exclusively of discrete and faithful representations. Learn about the groundbreaking work with Beyrer-Guichard-Labourie-Wienhard demonstrating that for classical groups G, these theories are explained by Θ-positivity, a Lie algebraic framework developed by Guichard-Wienhard. Discover how closedness in the character variety ultimately stems from a collar lemma that generalizes a fundamental geometric feature of hyperbolic surfaces, providing deep insights into the geometric and algebraic structures underlying these mathematical objects.