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Preprint Number 1133

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1133. Albert Garreta, Alexei Miasnikov, Denis Ovchinnikov
Random nilpotent groups, polycyclic presentations, and Diophantine problems

Submission date: 8 December 2016


We introduce a model of random f.g., torsion-free, 2-step nilpotent groups (in short, τ_2-groups). To do so, we show that these are precisely the groups that admit a presentation of the form ⟨ A, C | [a_i, a_j]= ∏_t c_t^{λ_{t,i,j}}}, (i< j), [A,C]=[C,C]=1 ⟩, where A= {a_1, ... , a_n }, and C={ c_1, \dots, c_m }. Hence, one may select a random τ_2-group G by fixing A and C, and then randomly choosing exponents λ_{t,i,j} with |λ_{t,i,j}| ≤ l, for some l.
We prove that, if m ≥ n-1 ≥ 1, then the following holds asymptotically almost surely, as l → ∞:
The ring of integers Z is e-definable in G, systems of equations over Z are reducible to systems over G (and hence they are undecidable), the maximal ring of scalars of G is Z, G is indecomposable as a direct product of non-abelian factors, and Z(G)=⟨ C ⟩. If, additionally, m ≤ n(n-1)/2, then G is regular (i.e. Z(G) ≤ 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.

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Full text arXiv 1612.02651: pdf, ps.

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