THE UNIVERSAL THEORY OF A FREE NILPOTENT GROUP: A PROGRESS REPORT

Abstract

Let $n$ and $c$ be nonnegative integers. A group satisfies $C T(n)$ provided the centralizer of any element omitted by the $n$th term of the upper central series is abelian. A result of Myasnikov and Remeslennikoy is that the universal theory of a free group $F$ is axiomatized by the quasi-identities true in $F$ together with $C T(0)$ when the models are restricted to $F$-groups. It was shown in [1] that if $G$ is free in the variety $\mathcal{N}_c$ of group nilpotent of class at most $c$, then $G$ satisfies $C T(c-1)$. Myasnikov posed the question of whether or not the universal theory of $G$ is axiomatized by the quasi-identities true in $G$ together with $C T(c-1)$ when the models are restricted to $G$-groups in the case $c=2$. The authors have shown in [4] that Myasnikov's question has a positive answer when $G$ has rank 2 . In this paper, we outline an approach to try to extend the result to arbitrary finite rank. Moreover, a counterexample is exhibited to show the analogous result is false already when $c=3$. A conjecture is made about what axioms may suffice when $c \geq 3$.

Received: July 7, 2023
Accepted: August 4, 2023