Random nilpotent groups, polycyclic presentations, and Diophantine problems

Albert Garreta; Alexei Miasnikov; Denis Ovchinnikov

doi:10.1515/gcc-2017-0007

Published by De Gruyter October 18, 2017

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Albert Garreta , Alexei Miasnikov and Denis Ovchinnikov

From the journal Groups Complexity Cryptology

https://doi.org/10.1515/gcc-2017-0007

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Abstract

We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ2-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〉, where A={a1,…,an} and C={c1,…,cm}. Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λt,i,j, with |λt,i,j|≤ℓ for some ℓ. We prove that if m≥n-1≥1, then the following hold asymptotically almost surely as ℓ→∞: the ring ℤ is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ, G is indecomposable as a direct product of non-abelian groups, and Z⁢(G)=〈C〉. We further study when Z⁢(G)≤Is⁡(G′). 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.

Keywords: Random groups; nilpotent groups; polycyclic groups; Diophantine problem

MSC 2010: 20F18; 20D15; 60G99

Funding source: FP7 Ideas: European Research Council

Award Identifier / Grant number: 336983

Funding source: Eusko Jaurlaritza

Award Identifier / Grant number: IT974-16

Funding statement: This work was supported in part by the ERC grant 336983 and by the Basque Government grant IT974-16.

References

[1] M. Cordes, M. Duchin, Y. Duong, M.-C. Ho and A. P. Sánchez, Random nilpotent groups I, preprint (2015), https://arxiv.org/abs/1506.01426. 10.1093/imrn/rnv370Search in Google Scholar

[2] K. Delp, T. Dymarz and A. Schaffer-Cohen, A matrix model for random nilpotent groups, preprint (2016), https://arxiv.org/abs/1602.01454. 10.1093/imrn/rnx128Search in Google Scholar

[3] M. Duchin, H. Liang and M. Shapiro, Equations in nilpotent groups, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4723–4731. 10.1090/proc/12630Search in Google Scholar

[4] A. Garreta, A. Miasnikov and D. Ovchinnikov, Properties of random nilpotent groups, preprint (2016), https://arxiv.org/abs/1612.01242. Search in Google Scholar

[5] M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146. 10.1007/s000390300002Search in Google Scholar

[6] W. Hodges, Model Theory, Encyclopedia Math. Appl. 42, Cambridge University Press, Cambridge, 1993. 10.1017/CBO9780511551574Search in Google Scholar

[7] D. F. Holt, B. Eick and E. A. O’Brien, Handbook of Computational Group Theory, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2005. 10.1201/9781420035216Search in Google Scholar

[8] Y. Ollivier, A January 2005 Invitation to Random Groups, Ensaios Mat. 10, Sociedade Brasileira de Matemática, Rio de Janeiro, 2005. 10.21711/217504322005/em101Search in Google Scholar

[9] V. A. Roman’kov, Universal theory of nilpotent groups, Mat. Zametki 25 (1979), no. 4, 487–495, 635. 10.1007/BF01688474Search in Google Scholar

[10] D. Segal, Polycyclic Groups, Cambridge Tracts in Math. 82, Cambridge University Press, Cambridge, 1983. 10.1017/CBO9780511565953Search in Google Scholar

Received: 2016-12-16

Published Online: 2017-10-18

Published in Print: 2017-11-1

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Abstract

References

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