Quantifying conjugacy separability in wreath products of groups

Ferov, Michal; Pengitore, Mark

doi:10.48550/arxiv.1910.02174

Cited by 3 publications

(12 citation statements)

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“…Similarly, if π : A → A is a surjective homomorphism, we denote π : A ≀ B → A ≀ B as the canonical extension of π to all of A≀B. We refer readers unfamiliar with the concept of a canonical extension to [10,Subsection 4.1] for the formal definition and more details.…”

Section: Lemma 27 Let a B Be Abelian Groups And Letmentioning

confidence: 99%

“…Conj G (n) n d2 . In [10], the authors of this note gave a general formula in terms of depth functions of the factors for upper bounds for conjugacy depth functions of wreath products of conjugacy separable groups, generalising Remeslennikov's classification of conjugacy separable wreath products [21]. However, when applied directly to wreath products of abelian groups, the formulas given in [10] produce rather rough upper bounds.…”

mentioning

confidence: 99%

“…In [10], the authors of this note gave a general formula in terms of depth functions of the factors for upper bounds for conjugacy depth functions of wreath products of conjugacy separable groups, generalising Remeslennikov's classification of conjugacy separable wreath products [21]. However, when applied directly to wreath products of abelian groups, the formulas given in [10] produce rather rough upper bounds. Applying [10,Theorem C] to the lamplighter group F 2 ≀ Z, one gets that that its conjugacy depth function can be bound from above by the function 2 n n 2 , and by an application of [10,Theorem C] to the group Z ≀ Z, one gets that its conjugacy depth function can be bounded from above by the function n n n 2…”

mentioning

confidence: 99%

“…However, when applied directly to wreath products of abelian groups, the formulas given in [10] produce rather rough upper bounds. Applying [10,Theorem C] to the lamplighter group F 2 ≀ Z, one gets that that its conjugacy depth function can be bound from above by the function 2 n n 2 , and by an application of [10,Theorem C] to the group Z ≀ Z, one gets that its conjugacy depth function can be bounded from above by the function n n n 2…”

mentioning

confidence: 99%

“…In this note, we extend the methods introduced in [11] to the setting of all the groups of the form A ≀ B, where both A and B are finitely generated abelian groups. This allows us to use more effective methods to obtain much better upper bounds than those presented in [10]. Additionally, we are able to use methods from commutative algebra to produce lower bounds.…”

mentioning

confidence: 99%

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Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

Ferov¹,

Pengitore²

2022

Preprint

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In this note, we complete the study of asymptotic behaviour of conjugacy separability of the general case of wreath products of finitely abelian groups where the base group is possibly infinite. In particular, we provide super-exponential upper and lower bounds for conjugacy separability of wreath products where the base group contains Z and, combining with previous work of the authors, we provide asymptotic bounds for conjugacy separability depth functions of all wreath products of finitely generated abelian groups. As an application, we give exponential lower bounds for infinitely many wreath products where the acting group is not necessarily abelian.

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Section: Lemma 27 Let a B Be Abelian Groups And Letmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

Ferov¹,

Pengitore²

2022

Preprint

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Bounding conjugacy depth functions for wreath products of finitely generated abelian groups

Ferov,

Pengitore

2023

Journal of Groups, Complexity, Cryptology

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In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.

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