Random nilpotent groups, polycyclic presentations, and Diophantine problems

TL;DR

This paper introduces a model for random torsion-free 2-step nilpotent groups, analyzing their algebraic properties and decidability aspects, and extends the approach to polycyclic and higher-step nilpotent groups.

Contribution

It provides a new probabilistic model for random nilpotent groups and establishes their typical algebraic and logical properties, including undecidability and definability results.

Findings

Ring of integers is e-definable in the groups

Systems of equations over integers reduce to those over the groups

Groups are typically indecomposable and have trivial maximal ring of scalars

Abstract

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid [a_i, a_j]= \prod_t {\scriptstyle c_t^{\scriptscriptstyle \lambda_{t,i,j}}} \ (i< j), \ [A,C]=[C,C]=1\rangle,$ where $A=\{a_1, \dots, a_n\}$, and $C=\{c_1, \dots, c_m\}$. Hence, one may select a random $\tau_2$-group $G$ by fixing $A$ and $C$, and then randomly choosing exponents $\lambda_{t,i,j}$ with $|\lambda_{t,i,j}|\leq \ell$, for some $\ell$. We prove that, if $m\geq n-1\geq 1$, then the following holds asymptotically almost surely, as $\ell\to \infty$: The ring of integers $\mathbb{Z}$ is e-definable in $G$, systems of equations over $\mathbb{Z}$ are reducible to systems over $G$ (and hence they are undecidable), the maximal ring of scalars of $G$ is $\mathbb{Z}$, $G$ is indecomposable as a direct product of non-abelian factors, and $Z(G)=\langle C \rangle$. If, additionally, $m \leq n(n-1)/2$, then $G$ is regular (i.e. $Z(G)\leq {\it Is}(G')$). This is not the case if $m > n(n-1)/2$. In the last section of the paper we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.