Compactifying Higher rank Teichmüller spaces

Maximal and Hitchin representations are discrete subgroups of semisimple Lie groups isomorphic to the fundamental group of a surface. Their parameter spaces are often referred to as Higher rank Teichmüller spaces: not only they form entire connected components of the character variety, but the subgroups they parametrize also share many common geometric properties with holonomies of hyperbolizations. After introducing these representations and motivating their study, I will discuss a project joint with Burger, Iozzi and Parreau aimed at compactifying these spaces by studying associated actions on affine buildings. The techniques we use range from geometric group theory to real algebraic geometry.