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

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and measure rigidity play important roles in various settings. First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity results. Then, using measure rigidity, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.