Hereditary conjugacy separability of free products with amalgamation

Chagas, S. C.; Zalesskiĭ, Pavel

doi:10.1016/j.jpaa.2012.08.011

Cited by 5 publications

(8 citation statements)

References 22 publications

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“…. , (m − 1)al, as is deduced from equation (7). Moreover, we claim that k i+bl = k r b i for i = 1, .…”

Section: Primitive Stability In Graph Productssupporting

confidence: 66%

“…. , bl, where thek j are the elements appearing in equation (8) and the k j are the (renaming of the) elements appearing in equation (7). By construction, as seen in equation (8), we have thatk i+bl =k r b i for i = 1, .…”

Section: Primitive Stability In Graph Productsmentioning

confidence: 99%

“…Allenby and Gregorac [1, Theorem 2'] (compare also Bobrovskii and Sokolov [4]) showed that the class of CSS groups is closed under amalgamation along a common retract. More recently, Chagas and Zaleskii [7,Theorem 1.4] showed that right-angled Artin groups and their virtual retracts have many separability properties, one of which implies being CSS.…”

Section: Introductionmentioning

confidence: 99%

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Separating cyclic subgroups in graph products of groups

Berlai

Ferov

2019

Journal of Algebra

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We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products.Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups -which are called p-isolated -are closed in the pro-p topology of the graph product. In particular, we show that every pisolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology, and we fully characterise such subgroups.

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“…. , (m − 1)al, as is deduced from equation (7). Moreover, we claim that k i+bl = k r b i for i = 1, .…”

Section: Primitive Stability In Graph Productssupporting

confidence: 66%

Section: Primitive Stability In Graph Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Separating cyclic subgroups in graph products of groups

Berlai

Ferov

2019

Journal of Algebra

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“…Since Fuchsian groups are conjugacy separable [FR90] we deduce that every-finite index subgroup of is conjugacy separable. Then by [Min12, Corollary 12.3], and by [CZ13, Lemma 2.3 combined with Theorem 2.14]. But , so as required.◻…”

Section: The Jsj Decompositionmentioning

confidence: 86%

Profinite detection of 3-manifold decompositions

Wilton¹,

Zalesskiĭ²

2019

Compositio Math.

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The profinite completion of the fundamental group of a closed, orientable 3-manifold determines the Kneser-Milnor decomposition. If M is irreducible, then the profinite completion determines the Jaco-Shalen-Johannson decomposition of M .When trying to distinguish two compact 3-manifolds M, N, in practice the easiest method is often to compute some finite quotients of their fundamental groups, and notice that there is a finite group Q which is a quotient of π 1 M, say, but not of π 1 N. It would be very useful, both theoretically and in practice, to know that this method always works. The set of finite quotients of a group Γ is encoded by the profinite completionΓ (the inverse limit of the system of finite quotient groups), and so one is naturally led to the following question.Question 0.1. Let M be a compact, orientable 3-manifold. To what extent is π 1 M determined by its profinite completion?In particular, if M is determined among all compact, orientable 3-manifolds by π 1 M, then M is said to be profinitely rigid. If there are at most finitely many compact, orientable 3-manifolds N with π 1 M ≅ π 1 N, then M is said

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“…Case 1: Suppose that g m is conjugate to an element of a parabolic subgroup of H. We may assume then that g m is in some parabolic subgroup P , and since parabolic subgroups are virtually malnormal (See [14 [26,Proposition 3.2], we may deduce that γ belongs to the closure of this parabolic subgroup in G. One deduces from the cyclic subgroup separability of virtually special groups (see [8,Theorem 1.4]) that the closure of a free abelian group in H is a free profinite abelian group, i.e. the closure coincides with its profinite completion.…”

Section: Conjugacy Separabilitymentioning

confidence: 99%