# Cup products, lower central series, and holonomy Lie algebras

## TL;DR

This paper extends the relationship between graded Lie algebras and holonomy Lie algebras to all finitely presented groups, providing explicit formulas and algorithms for computation, with applications in topology and group theory.

## Contribution

It generalizes key results to arbitrary finitely presented groups and introduces an explicit formula and an algorithm for computing holonomy Lie algebras.

## Key findings

- Explicit formula for cup-product in cohomology of finite 2-complex
- Algorithm for computing holonomy Lie algebra using Magnus expansion
- Applications to link groups, one-relator groups, and Seifert fibered manifolds

## Abstract

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.