Current Developments in Mathematics

Volume 2018

Recent progress in the Zimmer program

Pages: 1 – 56



Aaron Brown (University of Chicago, Illinois, U.S.A.)


This article surveys recent results due to the author and his collaborators on rigidity properties of actions of certain countably infinite groups on compact manifolds. We specifically focus on the results of [12–14, 16, 18]. We primarily focus on groups such as $\Gamma = \operatorname{SL}(n, \mathbb{Z})$ (for $n \geq 3$) and more general lattices $\Gamma$ in (typically higher-rank) semisimple Lie groups. The actions considered will be either on low-dimensional manifolds (where the dimension is small relative to certain algebraic data associated with the acting group) or actions on tori $\mathbb{T}^d$ and nil-manifolds $N / \Lambda$.

Published 17 December 2019