Ergodic quasi-isometries and the cohomology of nilpotent Lie groups

We give a simplified and self-contained proof of the following result due to Shalom, Sauer, and Gotfredsen-Kyed: two quasi-isometric simply connected nilpotent Lie groups G and H have isomorphic cohomology algebras. Our proof is based on considering maps which induce an ergodic measure on the space of functions from G to H (ergodic maps), and we show that, given an ergodic quasi-isometry, one can construct an explicit isomorphism from H^*(H) to H^*(G). Specifically, when ψ is an ergodic quasi-isometry, the pullback ψ^*ω of a differential form ω has a well-defined amenable average ψ^*̄ω, and we show that ψ^*̄ is the desired isomorphism. A key observation in our proof is that quasi-isometries of nilpotent groups are coarsely volume-preserving, so the amenable average of the pullback of the volume form is always nonzero.