# Random nilpotent groups, polycyclic presentations, and Diophantine problems

Random nilpotent groups, polycyclic presentations, and Diophantine problems 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

# Random nilpotent groups, polycyclic presentations, and Diophantine problems

, Volume 9 (2): 17 – Nov 1, 2017
17 pages

/lp/de-gruyter/random-nilpotent-groups-polycyclic-presentations-and-diophantine-itnphaANqn
Publisher
de Gruyter
© 2017 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2017-0007
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Groups Complexity Cryptologyde Gruyter

Published: Nov 1, 2017

### References

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