Appears in collection : 2025 IHES Summer School – Discrete Subgroups of Lie Groups : Dynamics, Actions, Rigidity

The study of diagonal group actions on homogeneous spaces occupies a central role in modern homogeneous dynamics, with deep connections to number theory, ergodic theory, and the geometry of locally symmetric spaces. In this talk, we focus on the ones which escape to infinity. We explore recent advances in the analysis of such orbits, emphasizing the use of representation theory and their geometric properties as powerful tools to probe their structure, and more specifically to distinguish between certain types of them.