An Introduction to Higher Rank Teichmüller Theory - 1/4

Explore discrete subgroups of semisimple Lie groups through this mathematical lecture that introduces higher rank Teichmüller theory. Delve into fundamental groups of surfaces and their rich deformation theory, examining how these groups can be parametrized as subsets of character varieties. Learn about the Anosov condition and its role in describing open subsets of character varieties, 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, which provides a Lie algebraic explanation for these phenomena. Understand how closedness in character varieties relates to collar lemmas, which generalize key geometric features from hyperbolic surfaces to higher rank settings. This presentation forms the first part of a comprehensive minicourse delivered by Maria Beatrice Pozzetti from Università di Bologna at the Institut des Hautes Etudes Scientifiques, combining advanced concepts from geometric group theory, Lie theory, and Teichmüller theory.