|Date||Mar 31, 2023|
We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in SLnR (n is at least 3). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most (n-1), the action is either isometric or projective. Both cases, we don’t have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed n-manifold by a lattice in SLnR (n is at least 3) then the lattice should be commensurable with SLnZ. This is the work in progress with Aaron Brown.