2024/05/28 by Gioacchino Antonelli, Robert Young, Antonelli, Gioacchino +2
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #math.GR #math.MG
paper · pdf · doi:10.48550/arxiv.2405.18598
openalex publication_date 2024/05/28 · openalex created_date 2024/05/31 · openalex updated_date 2026/07/28
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 ψ that induces an ergodic measure on the space of functions from G to H. We call such maps ergodic maps. We show that when ψ is an ergodic map, the pullback ψ^*ω of a differential form ω admits a well-defined amenable average ψ*ω, and ψ^* is a homomorphism of cohomology algebras. In the case that f is a quasi-isometry, the ergodic map ψ is also a quasi-isometry, and ψ^* 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.