Infinitesimal finiteness obstructions

TL;DR

This paper introduces a rational homotopy obstruction that determines when a space with good finiteness properties can be modeled algebraically with similar properties, revealing new insights into group theory and algebraic models.

Contribution

It provides a novel rational homotopy obstruction criterion and connects finiteness properties of groups with the structure of their Malcev Lie algebras.

Findings

Maximal metabelian quotient of a large finitely generated group is not finitely presented

A finitely generated group admits a connected 1-model with finite-dimensional degree 1 if its Malcev Lie algebra is a certain completion

The obstruction links topological finiteness properties to algebraic models via rational homotopy theory

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.