Diophantine problems in solvable groups

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

doi:10.1142/s1664360720500058

Cited by 17 publications

(8 citation statements)

References 48 publications

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“…Since G is verbally elliptic (i.e. all verbal subgroups of G have finite verbal width) [22], we have that G s is e-interpretable in G [16,Section 2.3]. Note also that G s is normal in G and that A " G{G s is a virtually polycyclic group that is periodic, i.e.…”

Section: Lemma 72 Let G Be a Virtually Polycyclic Group And Let H Be ...mentioning

confidence: 99%

Group equations with abelian predicates

Ciobanu¹,

Garreta²

2022

Preprint

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In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups.We use interpretability by equations to establish model-theoretic and algebraic conditions which are sufficient to get undecidability. We apply our results to (nonabelian) right-angled Artin groups, and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in rightangled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation.

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Section: Lemma 72 Let G Be a Virtually Polycyclic Group And Let H Be ...mentioning

confidence: 99%

Group equations with abelian predicates

Ciobanu¹,

Garreta²

2022

Preprint

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“…At present, there are known many rings, groups, and monoids R where the arithmetic N is interpretable (see, for instance, the Sections 6-14 below), so for all of them Corollary 1 holds when O or F is replaced by R. However, the rings O and F above most often appear as interpretable in other structures, especially, the rings of algebraic integers O typically occur via interpretations in finitely-generated groups (see, for example, [17,18,19], where the rings O are interpreted in groups and rings by means of equations). In this direction we would like to mention the following general result.…”

Section: Properties Preserved Under Interpretationsmentioning

confidence: 99%

Rich groups, weak second order logic, and applications

Kharlampovich¹,

Myasnikov²,

Sohrabi³

2021

Preprint

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In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.

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“…It is nothing else than the classical model-theoretic technique of interpretability (see [12,17]), restricted to systems of equations (equivalently, positive existential formulas without disjunctions). In [8,9] we used this technique to study the Diophantine problem in different classes of solvable groups and rings.…”

Section: Diophantine Problem and E-interpretabilitymentioning

confidence: 99%

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

Garreta¹,

Legarreta²,

Miasnikov³

et al. 2020

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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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Diophantine problems in solvable groups

Cited by 17 publications

References 48 publications

Group equations with abelian predicates

Group equations with abelian predicates

Rich groups, weak second order logic, and applications

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

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