An axiomatization for the universal theory of the Heisenberg group

Abstract

The Heisenberg group, here denoted $H$, is the group of all $3\times 3$ upper unitriangular matrices with entries in the ring $\mathbb{Z}$ of integers. A.G. Myasnikov posed the question of whether or not the universal theory of $H$, in the language of $H$, is axiomatized, when the models are restricted to $H$-groups, by the quasi-identities true in $H$ together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.