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 𝕋n, and try to present the ideas of the proof. The result imply existence of invariant measures for SL(n,ℤ) actions on 𝕋n with standard homotopy data as well as global rigidity of Anosov actions on infranilmanifolds and existence of semiconjugacies without assumption on existence of invariant measure.
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