Linking of letters and the lower central series of free groups

TL;DR

This paper introduces new invariants for the lower central series of free groups using linking of letters, providing a dual basis for free Lie algebras and connecting to Lie coalgebra theory.

Contribution

It develops a novel approach to invariants of the lower central series via linking of letters, extending Lie coalgebra methods and offering a new co-basis for free Lie algebras.

Findings

Invariants span the rational dual of lower central series subquotients

New co-basis for free Lie algebras established

Comparison with Fox derivatives enhances understanding

Abstract

We develop invariants of the lower central series of free groups through linking of letters, showing they span the rational linear dual of the lower central series subquotients. We build on an approach to Lie coalgebras through operads, setting the stage for generalization to the lower central series Lie algebra of any group. Our approach yields a new co-basis for free Lie algebras. We compare with the classical approach of Fox derivatives. v2: substantial refinement of exposition