On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

Darbinyan, Arman; Mikaelian, Vahagn H.

doi:10.48550/arxiv.1201.5505

Cited by 2 publications

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“…In the paper [12] (see also [5]) the following fact was proved Lemma 1. Let A and B be linearly ordered groups and Γ be a subgroup of…”

Section: Note That From the Above Definition It Follows Thatmentioning

confidence: 93%

Group Embeddings with Algorithmic Properties

Darbinyan

2015

Communications in Algebra

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We show that every countable group H with solvable word problem can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G. IntroductionIn the famous paper [8] by Higman, B.H.Neumann and H. Neumann in 1949, using constructions based on HNN-extensions, it was shown that every countable group can be embedded in a group generated by two elements. Later, B.H. Neumann and H. Neumann suggested an alternative embedding construction based on wreath products [15], which allowed them to show that every countable solvable group can be embedded in a 2-generated solvable group. Further development of these ideas was done by Hall [7]. Subsequently, other constructions based on this ideas were introduced, where the 2-generated group inherited some other properties of the initial group. Most of the aforementioned embedding constructions were motivated either by the desire to better control the algebraic structure or geometric properties of the embedding, but none of the constructions based on wreath products was

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“…In the paper [12] (see also [5]) the following fact was proved Lemma 1. Let A and B be linearly ordered groups and Γ be a subgroup of…”

Section: Note That From the Above Definition It Follows Thatmentioning

confidence: 93%

Group Embeddings with Algorithmic Properties

Darbinyan

2015

Communications in Algebra

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“…Among other recent research on embeddings of into 2-generator groups we would like to stress the following: [4] continues the linear order relation of rational numbers in onto the whole 2-generator group T , and it shows that the embedding can be verbal. [20] mentions that can never be embedded into a finitely generated metabelian group T (see Section 7 in [20] and also [6]).…”

Section: Introductionmentioning

confidence: 96%

Embeddings using universal words in the free group of rank 2

Mikaelian¹

2020

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For a countable group G = 〈 A | R 〉 presented by its generators A and defining relations R we discuss a simple method of embedding for G into such a 2-generator group T that the images of generators from A are explicitly given in T , and the defining relations for T can automatically be deduced from the relations R. The obtained method is applied on particular examples of groups, and references of its application for embeddings of recursive groups into finitely presented groups are given.

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On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

Cited by 2 publications

References 11 publications

Group Embeddings with Algorithmic Properties

Group Embeddings with Algorithmic Properties

Embeddings using universal words in the free group of rank 2

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