The space of characters, its dynamics, and applications to arithmetic groups

To any group $G$ is associated the space $\mathrm{Ch}(G)$ of all characters on $G$. After discussing this space, I'll turn to discuss dynamics on it, namely, invariant measures and random walks. The action of any arithmetic group on the character space of its amenable/solvable radical is stiff, i.e, any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness is used to deduce dichotomy statements on higher rank arithmetic groups pertaining normal subgroups, dynamical systems, representations and more.

The talks is based on a joint work with Uri Bader.