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

Garreta, Albert; Miasnikov, Alexei; Ovchinnikov, D. V.

doi:10.1016/j.jalgebra.2020.01.030

Cited by 7 publications

(8 citation statements)

References 13 publications

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“…In Section 3 we study the Diophantine problem in finitely generated metabelian groups given by full rank presentations. This is a continuation of research in [7] and [10].…”

Section: Diophantine Problemsmentioning

confidence: 60%

“…We showed in [10] that if a finitely generated nilpotent group G admits a full-rank presentation, then G is either virtually free nilpotent (provided the deficiency d ě 2), or virtually cyclic (if d " 1), or finite (if d ď 0).…”

Section: Metabelian Groups With Full Rank Presentationsmentioning

confidence: 99%

“…Groups given in the variety M of all metabelian groups by full rank presentations also have a rather restricted structure, as witnessed by the following result. In another direction, we showed in [10] that in any direct decomposition of a nilpotent group G given in the nilpotent variety N c , c ě 2, by a finite full rank presentation of deficiency ě 1, all, but one, direct factors are finite.…”

Section: Metabelian Groups With Full Rank Presentationsmentioning

confidence: 99%

“…For the variety of nilpotent groups N c (of a fixed nilpotency class c), this approach has been studied recently in [3,10]. Other models of randomness for the groups in N c can be found in [5,7].…”

Section: Random Finitely Presented Groups: the Few-relations Modelmentioning

confidence: 99%

“…Theorem 1.6. [10] One of the main appeals of full-rank presentations is that they occur asymptotically almost surely in the few-relators model for random groups in any variety V, in particular, in the variety M. Hence, random metabelian groups (in the few relators model) have full rank presentations asymptotically almost surely. Therefore, all the properties described above for metabelian groups given by full rank presentations are generic in the class of finitely presented metabelian groups.…”

Section: Random Finitely Presented Groups: the Few-relations Modelmentioning

confidence: 99%

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

Garreta¹,

Legarreta²,

Miasnikov³

et al. 2020

Preprint

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We study metabelian groups G given by full rank finite presentations xA | Ry M in the variety M of metabelian groups. We prove that G is a product of a free metabelian subgroup of rank maxt0, |A| ´|R|u and a virtually abelian normal subgroup, and that if |R| ď |A| ´2 then the Diophantine problem of G is undecidable, while it is decidable if |R| ě |A|. We further prove that if |R| ď |A| ´1 then in any direct decomposition of G all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically almost surely, finitely presented metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

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“…In Section 3 we study the Diophantine problem in finitely generated metabelian groups given by full rank presentations. This is a continuation of research in [7] and [10].…”

Section: Diophantine Problemsmentioning

confidence: 60%

Section: Metabelian Groups With Full Rank Presentationsmentioning

confidence: 99%

Section: Metabelian Groups With Full Rank Presentationsmentioning

confidence: 99%

Section: Random Finitely Presented Groups: the Few-relations Modelmentioning

confidence: 99%

Section: Random Finitely Presented Groups: the Few-relations Modelmentioning

confidence: 99%

See 3 more Smart Citations

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

Garreta¹,

Legarreta²,

Miasnikov³

et al. 2020

Preprint

Self Cite

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show abstract

The Diophantine problem in Chevalley groups

Bunina,

Myasnikov,

Plotkin

2024

Journal of Algebra

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The Diophantine problem in the classical matrix groups

Myasnikov¹,

Sohrabi²

2021

Izv. Math.

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In this paper we study the Diophantine problem in the classical matrix groups , , and , , over an associative ring with identity. We show that if is one of these groups, then the Diophantine problem in is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in . When we assume that is commutative. Similar results hold for and provided has no zero divisors (for the ring is not assumed to be commutative).

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

Cited by 7 publications

References 13 publications

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

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

The Diophantine problem in Chevalley groups

The Diophantine problem in the classical matrix groups

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