A Matrix Model for Random Nilpotent Groups

Delp, Kelly; Dymarz, Tullia; Schaffer-Cohen, Anschel

doi:10.1093/imrn/rnx128

Cited by 3 publications

(2 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in Lemma 5. 2 we show that such model yields finite groups asymptotically almost surely. This is the reason why we have particularized it to τ 2 -groups.…”

Section: Introductionmentioning

confidence: 74%

See 1 more Smart Citation

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Garreta

Miasnikov

Ovchinnikov

2017

Groups Complexity Cryptology

View full text Add to dashboard Cite

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 xA, C | rai, ajs " ś t c λ t,i,j t pi ă jq, rA, Cs " rC, Cs " 1y, where A " ta1, . . . , anu, and C " tc1, . . . , cmu. 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| ď ℓ, for some ℓ.We prove that, if m ě n´1 ě 1, then the following holds asymptotically almost surely, as ℓ Ñ 8: 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 ZpGq " xCy. If, additionally, m ď npn´1q{2, then G is regular (i.e. ZpGq ď IspG 1 q). This is not the case if m ą npn´1q{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.

show abstract

“…However, in Lemma 5. 2 we show that such model yields finite groups asymptotically almost surely. This is the reason why we have particularized it to τ 2 -groups.…”

Section: Introductionmentioning

confidence: 74%

“…These words are added as relators to N s,m , yielding a random f.g. nilpotent group G " N s,m {xxRyy. An alternative model was introduced in [2], where f.g. torsion-free nilpotent groups are considered as subgroups of unitriangular matrices.…”

Section: Introductionmentioning

confidence: 99%

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Garreta

Miasnikov

Ovchinnikov

2017

Groups Complexity Cryptology

View full text Add to dashboard Cite

show abstract

Full rank presentations and nilpotent groups: Structure, Diophantine problem, and genericity

2020

View full text Add to dashboard Cite

Metabelian groups: Full-rank presentations, randomness and Diophantine problems

Garreta

Legarreta

Miasnikov

et al. 2020

Journal of Group Theory

View full text Add to dashboard Cite

We study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

show abstract

A Matrix Model for Random Nilpotent Groups

Cited by 3 publications

References 4 publications

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Full rank presentations and nilpotent groups: Structure, Diophantine problem, and genericity

Metabelian groups: Full-rank presentations, randomness and Diophantine problems

Contact Info

Product

Resources

About