Full rank presentations and nilpotent groups: structure, Diophantine problem, and genericity

We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency $|A| - |R|$ is at least $2$ then the group $G$ is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations $\langle A \mid R\rangle$ is undecidable if $|A| - |R| \geq 2$ and decidable otherwise. We show that this class of groups is rather large since finite presentations asymptotically almost surely have full rank, so a random nilpotent group in the few relators model has a full rank presentation asymptotically almost surely. Full rank presentations give one a useful tool to approach random nilpotent groups and study their properties. Note, that the results above significantly improve our understanding of the Diophantine problem in finitely generated nilpotent groups: from a few special examples of groups with undecidable Diophantine problem we got to the place where we know that the Diophantine problem in all "typical" nilpotent groups is also undecidable.