Submission date: 5 December 2016

Abstract:

We study random nilpotent groups of the form G=N/⟨⟨ R ⟩⟩, where N is a non-abelian free nilpotent group with m generators, and R is a set of r random relators of length l. We prove that the following holds asymptotically almost surely as l → ∞:
1) If r ≤ m-2, then the ring of integers Z is e-definable in G/ Is(G_3), and systems of equations over Z are reducible to systems of equations over G (hence, they are undecidable). Moreover, Z(G) ≤ 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 ≥ m+1, then G is finite and abelian.

In the last three cases, systems of equations are decidable in G.

Mathematics Subject Classification:

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Full text arXiv 1612.01242: pdf, ps.