Appears in collection : 2025 - T2 - Higher rank geometric structures

This minicourse will delve into the theory of cocompact domains of discontinuity for Anosov representations, with a particular focus on their construction and properties. We will introduce the necessary background on flag varieties, Lie theory, and Anosov representations. We will then explore the foundational work of Guichard and Wienhard, and, time permitting, also the work of Kapovich-Leeb-Porti. We will demonstrate that the spaces constructed are indeed cocompact domains of discontinuity. The course will then highlight joint work with Alessandrini, Tholozan, and Wienhard, showing that, in many cases, the quotient of these domains by the Anosov representation is homeomorphic to a fiber bundle. Finally, we will examine the topology of the fiber in some specific cases. The course will feature guest lectures by León Carvajales, who will discuss his joint work with Stecker on non-cocompact domains of discontinuity on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces, and Mason Hart, who will present his research on the topology of the fiber using Fintushel's classification of circle actions on 4-manifolds.