Properties of random nilpotent groups

Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov

Published 2016-12-05Version 1

We study random nilpotent groups of the form $G=N/\langle\langle R \rangle \rangle$, where $N$ is a non-abelian free nilpotent group with $m$ generators, and $R$ is a set of $r$ random relators of length $\ell$. We prove that the following holds asymptotically almost surely as $\ell\to \infty$: 1) If $r\leq m-2$, then the ring of integers $\mathbb{Z}$ is e-definable in $G/{it Is}(G_3)$, and systems of equations over $\mathbb{Z}$ are reducible to systems of equations over $G$ (hence, they are undecidable). Moreover, $Z(G)\leq {\it Is}(G')$, $G/G_3$ is virtually free nilpotent of rank $m-r$, and $G/G_3$ cannot be decomposed as the direct product of two non-virtually abelian groups. 2) If $r=m-1$, then $G$ is virtually abelian. 3) If $r= m$, then $G$ is finite. 4) If $r\geq m+1$, then $G$ is finite and abelian. In the last three cases, systems of equations are decidable in $G$.