Shirshov's amalgamated free product and generic nilpotent groups
In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the \textit{cofinal} amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the \textit{weak} amalgamation property). Second, the reduct of the Fra\"iss\'e limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.