On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Sokolov, E. V.

doi:10.1142/s0218196714500313

Cited by 10 publications

(6 citation statements)

References 10 publications

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“…In [Log00], the author gives a criterion under which free products with commuting subgroups are cyclic subgroup separable. This was later generalized in [Sok14]. This criterion will be used throughout this section; as a warm-up we prove the following lemma which easily follows from Theorem 3.2: Lemma 3.3.…”

Section: A-completions and Cyclic Subgroup Separabilitymentioning

confidence: 97%

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

Fruchter¹

2021

Preprint

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We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over coherent RAAGs and toral relatively hyperbolic groups have a solvable word problem.

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Section: A-completions and Cyclic Subgroup Separabilitymentioning

confidence: 97%

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

Fruchter¹

2021

Preprint

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“…22]) that if A, B are two groups with H ≤ A and K ≤ B, then the group Gp A, B | [H, K] = 1 is called the free product of A and B with commuting subgroups H and K. Such products have seen some recent study, see e.g. [78].…”

Section: B(γmentioning

confidence: 99%

On the Diophantine problem in some one-relator groups

Nyberg-Brodda¹

2022

Preprint

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We study the Diophantine problem, i.e. the decision problem of solving systems of equations, for some families of one-relator groups, and provide some background for why this problem is of interest. The method used is primarily the Reidemeister-Schreier method, together with general recent results by Dahmani & Guirardel and Ciobanu, Holt & Rees on the decidability of the Diophantine problem in general classes of groups. First, we give a sample of the methods of the article by proving that the one-relator group with defining relation a m b n = 1 is virtually a direct product of hyperbolic groups for all m, n ≥ 0, and thus conclude decidability of the Diophantine problem in such groups. As a corollary, we obtain that the Diophantine problem is decidable in any torus knot group. Second, we study the twogenerator, one-relator groups Gm,n with defining relation a commutator [a m , b n ] = 1, where m, n ≥ 1. In doing so, we define and study a natural class of groups (rabsags), related to right-angled Artin groups (raags). We reduce the Diophantine problem in the groups Gm,n to the Diophantine problem in groups which are virtually certain rabsags. As a corollary of our methods, we show that the submonoid membership problem is undecidable in the group G 2,2 with the single defining relation [a 2 , b 2 ] = 1. We use the recent classification by Gray & Howie of raag subgroups of one-relator groups to classify the raag subgroups of some rabsags, showing the potential usefulness of one-relator theory to this area. Finally, we define and study Newman groups NG(p, q), which are (p + 1)-generated one-relator groups generalising the solvable Baumslag-Solitar groups. We show that all such groups are hyperbolic, and thereby also conclude decidability of their Diophantine problem.

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“…Groups satisfying the separability property with respect to the collection of all cyclic subgroups are known in the literature as cyclic subgroup separable groups or Π c groups. Such groups have received considerable attention; see for instance [2,18,29,30].…”

Section: Schützenberger Groupsmentioning

confidence: 99%

On separability finiteness conditions in semigroups

Miller¹,

O’Reilly²,

Quick³

et al. 2020

Preprint

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Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H-class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H-classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

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On the cyclic subgroup separability of the free product of two groups with commuting subgroups

Cited by 10 publications

References 10 publications

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

On the Diophantine problem in some one-relator groups

On separability finiteness conditions in semigroups

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