Model-theoretic properties of nilpotent groups and Lie algebras

Abstract

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent $p$ studied by Baudisch is 2-dependent and NSOP${}_1$. We prove that the class of $c$-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for $2<c$, the generic $c$-nilpotent Lie algebra over ${\mathbb{F}}_p$ is strictly NSOP${}_4$ and $c$-dependent. Via the Lazard correspondence, we obtain the same result for $c$-nilpotent groups of exponent $p$, for an odd prime $p>c$.

45 pages