Uniformity in Higher class Free Lie algebras

Abstract

Let ${\mathfrak{f}}_{c,2}$ denote a free class-$c$ Lie rings on $2$ generators. We investigate the zeta functions enumerating graded ideals in ${\mathfrak{f}}_{c,2}({\mathbb{F}}_p)$ for $c\le 6$, prove that they are uniformly given by polynomials in $p$ for $c\le 5$ and not uniformly given by a polynomial in $p$ for $c=6$. We also show that the zeta functions enumerating one-step graded ideals ${\mathfrak{f}}_{c,2}({\mathbb{F}}_p)$ is always given by a polynomial in $p$ for all $c$.

34 pages