Abstract
We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ2{\tau_{2}}-groups). To do so, we show that these are precisely the groups with presentation of the form 〈A,C∣[ai,aj]=∏t=1mctλt,i,j(1≤i<j≤n),[A,C]=[C,C]=1〉{\langle A,C\mid[a_{i},a_{j}]=\prod_{t=1}^{m}c_{t}^{\lambda_{t,i,j}}(1\leq i<j% \leq n),\,[A,C]=[C,C]=1\rangle}, where A={a1,…,an}{A=\{a_{1},\dots,a_{n}\}} and C={c1,…,cm}{C=\{c_{1},\dots,c_{m}\}}. Hence, a random G can be selected by fixing A and C , and then randomly choosing integers λt,i,j{\lambda_{t,i,j}}, with |λt,i,j|≤ℓ{|\lambda_{t,i,j}|\leq\ell} for some ℓ{\ell}. We prove that if m≥n-1≥1{m\geq n-1\geq 1}, then the following hold asymptotically almost surely as ℓ→∞{\ell\to\infty}: the ring ℤ{\mathbb{Z}} is e-definable in G , the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ{\mathbb{Z}}, G is indecomposable as a direct product of non-abelian groups, and Z(G)=〈C〉{Z(G)=\langle C\rangle}. We further study when Z(G)≤Is(G′){Z(G)\leq\operatorname{Is}(G^{\prime})}. Finally, 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.