In this paper, we study how the cohomology of nilpotent groups is affected by Lipschitz maps. We show that, given a smooth Lipschitz map f between two simply-connected nilpotent Lie groups G and H, there is a map \(\psi \) that induces an ergodic measure on the space of functions from G to H. We call such maps ergodic maps. We show that when \(\psi \) is an ergodic map, the pullback \(\psi ^*\omega \) of a differential form \(\omega \) admits a well-defined amenable average \(\overline{\psi ^{*}}\omega \) , and \(\overline{\psi ^*}\) is a homomorphism of cohomology algebras. In the case that f is a quasi-isometry, the ergodic map \(\psi \) is also a quasi-isometry, and \(\overline{\psi ^*}\) is an isomorphism. This lets us generalize and provide a simplified, self-contained proof of the theorem due to Shalom, Sauer, and Gotfredsen–Kyed that quasi-isometric nilpotent groups have isomorphic cohomology algebras.
