Ergodic maps and the cohomology of nilpotent Lie groups

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.