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 NSOP₁. 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 $F_p$ is strictly NSOP₄ and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime $p>c$.