Elementary coordinatization of finitely generated nilpotent groups

Myasnikov, A. G.; Sohrabi, Mahmood

doi:10.48550/arxiv.1311.1391

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Combining our results with the elementary classification of finitely generated nilpotent (R)-groups (see [34,31,6]), we deduce the following Theorem 10.13 (Rigidity of the class of graph products of (non-cyclic) fg nilpotent (well-structured nilpotent R-) groups). Let C n be the class of (non-cyclic) finitely generated nilpotent groups.…”

Section: After a Possible Reordering Of Factors)mentioning

confidence: 70%

On the elementary theory of graph products of groups

Casals-Ruiz,

Kazachkov,

de la Nuez González

2024

Dissertationes Math.

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In this paper, we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups, we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition that generalizes a range of natural "non-freeness conditions" such as the satisfaction of a group law, having a non-trivial center, or being boundedly simple.As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups and describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results for the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent, or classical linear groups.

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Section: After a Possible Reordering Of Factors)mentioning

confidence: 70%

On the elementary theory of graph products of groups

Casals-Ruiz,

Kazachkov,

de la Nuez González

2024

Dissertationes Math.

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“…Combining our results with the elementary classification of finitely generated nilpotent (R)groups, see [29,26,6], we deduce the following Theorem 10.13 (Rigidity of the class of graph products of (non-cyclic) fg nilpotent (well-structured nilpotent R-) groups).…”

Section: Applications 101 Elementary Equivalence Of Graph Products Of...mentioning

confidence: 73%

On the elementary theory of graph products of groups

Casals-Ruiz¹,

Kazachkov²,

González³

2021

Preprint

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In this paper we study the elementary theory of graph products of groups and show that under natural conditions on the vertex groups we can recover (the core of) the underlying graph and the associated vertex groups. More precisely, we require the vertex groups to satisfy a non-generic almost positive sentence, a condition which generalizes a range of natural "non-freeness conditions" such as the satisfaction of a group law, having nontrivial center or being boundedly simple.As a corollary, we determine an invariant of the elementary theory of a right-angled Artin group, the core of the defining graph, which we conjecture to determine the elementary class of the RAAG. We further combine our results with the results of Sela on free products of groups to describe all finitely generated groups elementarily equivalent to certain RAAGs. We also deduce rigidity results on the elementary classification of graph products of groups for specific types of vertex groups, such as finite, nilpotent or classical linear groups.

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“…We note that the existence of such functions extends to nilpotent groups admitting exponents in a binomial principal ideal domain, as described in Theorem 5.7 of [27].…”

Section: Proposition 21mentioning

confidence: 83%

Logspace and compressed-word computations in nilpotent groups

Macdonald¹,

Myasnikov²,

Николаев³

et al. 2015

Preprint

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For finitely generated nilpotent groups, we employ Mal'cev coordinates to solve several classical algorithmic problems efficiently. Computation of normal forms, the membership problem, the conjugacy problem, and computation of presentations for subgroups are solved using only logarithmic space and quasilinear time. Logarithmic space presentation-uniform versions of these algorithms are provided. Compressed-word versions of the same problems, in which each input word is provided as a straight-line program, are solved in polynomial time.

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Elementary coordinatization of finitely generated nilpotent groups

Cited by 4 publications

References 13 publications

On the elementary theory of graph products of groups

On the elementary theory of graph products of groups

On the elementary theory of graph products of groups

Logspace and compressed-word computations in nilpotent groups

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