Rigidity of hyperbolic higher rank lattice actions

I will discuss some recent results with Aaron Brown and Zhiren Wang on actions by higher rank lattices on nilmanifolds. I will present the result in the simplest case possible, SL(n,Z) acting on T^n, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,Z) actions on T^n with standard homotopy data as well as global rigidity of Ansosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.