Infinitesimal finiteness obstructions

Abstract

Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1-minimal models, we also show that a finitely generated group $\pi$ admits a connected 1-model with finite-dimensional degree 1 piece if and only if the Malcev Lie algebra $\mathfrak{m}(\pi)$ is the lower central series completion of a finitely presented Lie algebra.