Homology, lower central series, and hyperplane arrangements

TL;DR

This paper investigates the algebraic structures of finitely generated groups through nilpotent towers and Lie algebras, linking topological extension problems to hyperplane arrangement combinatorics.

Contribution

It generalizes Rybnikov's result by connecting isomorphism extension problems to hyperplane arrangements and shows nilpotent quotients are combinatorially determined.

Findings

Nilpotent quotients of decomposable arrangement groups are combinatorially determined.

Recasting Rybnikov's result within a broader algebraic framework.

Establishing links between group extensions and hyperplane arrangement topology.

Abstract

We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.