Ergodic maps and the cohomology of nilpotent Lie groups

TL;DR

This paper introduces ergodic maps between nilpotent Lie groups and demonstrates how they induce cohomology algebra isomorphisms, simplifying proofs of cohomological invariance under quasi-isometries.

Contribution

It defines ergodic maps and shows they induce cohomology algebra homomorphisms, extending to isomorphisms for quasi-isometries, thus simplifying existing proofs.

Findings

Ergodic maps induce well-defined cohomology homomorphisms.

Quasi-isometries induce cohomology algebra isomorphisms.

Provides a simplified proof of cohomological invariance for nilpotent groups.

Abstract

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.