Random nilpotent groups, polycyclic presentations, and Diophantine problems

We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ2-groups). To do so, we show that these are precisely the groups with presentation of the form (A, C | [ai , aj] = A^m_t=1 c^λt,i,j_t (1 ≤ i < j ≤ n), [A, C] = [C, C] = 1>,where A = {a1, . . . , an} and C = {c1, . . . , cm}. Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λ_t,i,j, with |λ_t,i,j| ≤ l for some l.We prove that if m ≥ n - 1 ≥ 1, then the following hold asymptotically almost surely as l → ∞: the ring Z is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is Z, G is indecomposable as a direct product of non-abelian groups, and Z(G) = (C).We further study when Z(G) ≤ Is(G'). 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.