2016/12/08 by Albert Garreta, Garreta, Albert, Alexei Miasnikov +3 · 1 citation
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR
paper · pdf · doi:10.48550/arxiv.1612.02651
23 pages. arXiv admin note: text overlap with arXiv:1612.01242
arxiv created 2016/12/08 · openalex publication_date 2016/12/08 · arxiv updated 2016/12/09 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
We introduce a model of random f.g., torsion-free, 2-step nilpotent groups (in short, τ2-groups). To do so, we show that these are precisely the groups that admit a presentation of the form ⟨ A, C | [ai, aj]= ∏t \scriptstyle ct^\scriptscriptstyle λt,i,j (i< j), [A,C]=[C,C]=1⟩, where A=\a1, …, an\, and C=\c1, …, cm\. Hence, one may select a random τ2-group G by fixing A and C, and then randomly choosing exponents λt,i,j with |λt,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)≤ \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.