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AbstractWe 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.
Groups Complexity Cryptology – de Gruyter
Published: Nov 1, 2017
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