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Separating cyclic subgroups in graph products of groups. (English) Zbl 1467.20031

Summary: 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 \(p\)-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-\(p\) topology, and we fully characterise such subgroups.

MSC:

20F65 Geometric group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E07 Subgroup theorems; subgroup growth
20F36 Braid groups; Artin groups
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References:

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