Lie powers of relation modules for groups

Abstract

Motivated by applications to abstract group theory, we study Lie powers of relation modules. The relation module associated to a free presentation $G=F/N$ of a group G is the abelianization $N_{ab}=N/[N,N]$ of N, with G-action given by conjugation in F. The degree n Lie power is the homogeneous component of degree n in the free Lie ring on $N_{ab}$ (equivalently, it is the relevant quotient of the lower central series of N). We show that after reduction modulo a prime p this becomes a projective G-module, provided $n>1$ and n is not divisible by p.