Title: Smooth actions on manifold by higher rank lattices
Abstract: We will discuss about smooth actions on manifolds by higher rank groups, such as lattices in SL(n, ℝ) with n ≥ 3 or ℤk with k ≥ 2.
The higher rank property of the acting group suggests that the actions are rigid, which means that the action should have an algebraic origin, such as the Zimmer program and the Katok-Spatzier conjecture. One of the main topics is about how we can give an algebraic structure on the acting space which is a smooth manifold.
We survey some of recent breakthroughs and then focus mainly on actions of higher rank lattices. In particular, we focus on actions on manifolds with “positive entropy” by lattices in SL(n, ℝ), n ≥ 3. When the manifold has dimension n, then we will see that the lattice is commensurable to SL(n, Z) from a certain “algebraic structure” on M coming from the dynamics.
Part of the talk is ongoing work with Aaron Brown.