Appears in collection : 2024 - T2 - WS2 - Group actions with hyperbolicity and measure rigidity

Discrete subgroups of $\mathrm{PSL}(2,\mathbb{C})$ are called Kleinian groups. After a brief review of the Mostow-Sullivan rigidity theorem, we will discuss a new rigidity theorem for general Kleinian groups and its various implications including a cross-ratio rigidity theorem. For convex cocompact Kleinian groups, we also present a measure-theoretic rigidity theorem. We will discuss how we use the dynamics of one-parameter diagonal flows on higher rank homogeneous spaces of infinite volume to prove these results.

(This talk is based on joint work with Dongryul Kim).