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.

24 pages; accepted for publication in the Journal of the London Mathematical Society

References in corpus (1)

• Toric complexes and Artin kernels

Cited by in corpus (2)

• Residually finite rationally $p$ groups
• Cohomology jump loci of 3-manifolds