Virtually free finite-normal-subgroup-free groups are strongly verbally closed

Klyachko, Anton A.; Mazhuga, Andrey M.; Miroshnichenko, Veronika Yu.

doi:10.1016/j.jalgebra.2018.05.028

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…Let N c be the variety of all nilpotent groups of class at most c and N r,c a free nilpotent group of finite rank r and nilpotency class c. A subgroup H of N r,c is verbally closed in N r,c if and only if H is a free factor of N r,c in the variety N c (equivalently, an algebraically closed subgroup, or a retract of N r,c ). Some other results on verbal closedness can be found in [30,31,37,38,39]. Problem 5.2 is solved in this paper, see Corollary 2.7.…”

Section: Resultsmentioning

confidence: 64%

“…In the second direction one studies properties of algebraically, existentially, and verbally closed groups in certain overgroups or classes of groups (see Definition 2.1 below and a general definition of S-closedness suggested by Neumann in [45]). For problems and results in this area see the surveys of Leinen [34] and Roman'kov [55], and the papers [4,30,31,36,42,44,45,56,57,58,60]. Note that this branch of group theory is closely related to logic in the form of model theory and recursive functions, see the book of Higman and Scott [22], appendix A.4 in the book of Hodges [23], and the paper [26].…”

Section: Introductionmentioning

confidence: 99%

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Equations in acylindrically hyperbolic groups and verbal closedness

Bogopolski¹

2018

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We describe solutions of the equation x n y m = a n b m in acylindrically hyperbolic groups (AH-groups), where a, b are non-commensurable special loxodromic elements and n, m are integers with sufficiently large common divisor. Using this description and certain test words in AHgroups, we study the verbal closedness of AH-subgroups in groups.A subgroup H of a group G is called verbally closed if for any word w(x 1 , . . . , x n ) in variables x 1 , . . . , x n and any element h ∈ H, the equation w(x 1 , . . . , x n ) = h has a solution in G if and only if it has a solution in H.Main Theorem: Suppose that G is a finitely presented group and H is a finitely generated acylindrically hyperbolic subgroup of G such that H does not have nontrivial finite normal subgroups. Then H is verbally closed in G if and only if H is a retract of G.The condition that G is finitely presented and H is finitely generated can be replaced by the condition that G is finitely generated over H and H is equationally noetherian.As a corollary, we solve Problem 5.2 from the paper [42] of Myasnikov and Roman'kov: Verbally closed subgroups of torsion-free hyperbolic groups are retracts. ContentsOLEG BOGOPOLSKI 11. A special case of the first main theorem 57 12. Test words in acylindrically hyperbolic groups 62 13. Verbal closedness for finitely generated acylindrically hyperbolic subgroups 64 14. Verbal closedness for equationally noetherian acylindrically hyperbolic subgroups 69 15. Appendix 72 References 76

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Section: Resultsmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 99%

Equations in acylindrically hyperbolic groups and verbal closedness

Bogopolski¹

2018

Preprint

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“…In [10] Mazhuga showed that if H is a verbally closed subgroup of a group G, then H is algebraically closed in G. This result (except the very special case H = Z 2 * Z 2 , which was first considered in [7]) follows from our Theorem C. Indeed, H can be splitted as H = A * B, where A and B are nontrivial; hence H is relatively hyperbolic with respect to {A, B}. It is well known that if a non-(virtually cyclic) group is relatively hyperbolic with respect to a collection of proper subgroups, then it is acylindrically hyperbolic.…”

Section: Proof Of Theorem Cmentioning

confidence: 96%

“…• Theorem C, says that for any clean acylindrically hyperbolic group H and any overgroup G of H the notions of verbal and algebraic closedness of H in G are equivalent. Some special cases of this theorem were considered earlier in [7] and [10], see Remark 7.1 below.…”

Section: Introductionmentioning

confidence: 95%

On finite systems of equations in acylindrically hyperbolic groups

Bogopolski¹

2019

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Let H be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system S of equations with constants from H is equivalent to a single equation.We also show that the algebraic set associated with S is, up to conjugacy, a projection of the algebraic set associated with a single splitted equation (such equation has the form w(x 1 , . . . , x n ) = h, where w ∈ F (X), h ∈ H).From this we deduce the following statement: Let G be an arbitrary overgroup of the above group H. Then H is verbally closed in G if and only if it is algebraically closed in G.Another corollary: If H is a non-cyclic torsion-free hyperbolic group, then every (possibly infinite) system of equations with finitely many variables and with constants from H is equivalent to a single equation.

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“…Заметим, что первый пункт приведенной выше теоремы описывает определенный класс сильно вербально замкнутых групп (в частности, все нетривиальные свободные группы принадлежат этому классу). Из результатов работ [11] и [13] следует:…”

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Свободные Произведения Групп Сильно Вербально Замкнуты

Мажуга¹,

Mazhuga

2019

Математический сборник

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1 Механико-математический факультет МГУ им. М.В.Ломоносова, аспирант 1 Всероссийская академия внешней торговли, младший научный сотрудник Института макроэкономических исследований УДК 512.543.72Аннотация В ряде недавних работ было установлено, что многие почти свободные группы, фундаментальные группы почти всех связных поверхностей и все группы, являющиеся нетривиальными свободными произведениями групп с тождествами, алгебраически замкнуты в любой группе, в которой они вербально замкнуты. В данной работе мы установим, что любая группа, являющаяся нетривиальным свободным произведением групп, алгебраически замкнута в любой группе, в которой она вербально замкнута.Ключевые слова: вербально замкнутые подгруппы, алгебраически замкнутые подгруппы, ретракты групп.

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Virtually free finite-normal-subgroup-free groups are strongly verbally closed

Cited by 10 publications

References 11 publications

Equations in acylindrically hyperbolic groups and verbal closedness

Equations in acylindrically hyperbolic groups and verbal closedness

On finite systems of equations in acylindrically hyperbolic groups

Свободные Произведения Групп Сильно Вербально Замкнуты

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