The Zimmer programme asks how lattices in higher-rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SLn(ℝ) with n ≥ 3 or other higher rank ℝ-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank ℝ-split semisimple Lie groups. For example, non-trivial actions by lattices in SLn(ℝ), n ≥ 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on ℝPn-1 up to a finite quotient group and a finite covering.
This is a joint work with Aaron Brown and Federico Rodriguez Hertz.
This video is part of the Institute for Advanced Study‘s Members’ colloquium.
