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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

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Publisher
de Gruyter
Copyright
© 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 Cryptology – de Gruyter

Published: Nov 1, 2017

References

  • Random walk in random groups