On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911879036665856 |
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| author | Sokolov, E. V. |
| author_facet | Sokolov, E. V. |
| contents | Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism $σ$ of $X$ onto a group from $\mathcal{C}$ such that the elements $xσ$ and $yσ$ are still not equal (respectively, not conjugate in $Xσ$). A generalized Baumslag-Solitar group or GBS-group is the fundamental group of a finite connected graph of groups whose all vertex and edge groups are infinite cyclic. An ordinary Baumslag-Solitar group is the GBS-group that corresponds to a graph containing only one vertex and one loop. Suppose that the class $\mathcal{C}$ consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy $\mathcal{C}$-separable if and only if it is residually a $\mathcal{C}$-group. We also find a criterion for a solvable GBS-group to be conjugacy $\mathcal{C}$-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09736 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups Sokolov, E. V. Group Theory 20E26, 20E08 (Primary) 20E06 (Secondary) Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism $σ$ of $X$ onto a group from $\mathcal{C}$ such that the elements $xσ$ and $yσ$ are still not equal (respectively, not conjugate in $Xσ$). A generalized Baumslag-Solitar group or GBS-group is the fundamental group of a finite connected graph of groups whose all vertex and edge groups are infinite cyclic. An ordinary Baumslag-Solitar group is the GBS-group that corresponds to a graph containing only one vertex and one loop. Suppose that the class $\mathcal{C}$ consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy $\mathcal{C}$-separable if and only if it is residually a $\mathcal{C}$-group. We also find a criterion for a solvable GBS-group to be conjugacy $\mathcal{C}$-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite. |
| title | On the conjugacy separability of ordinary and generalized Baumslag-Solitar groups |
| topic | Group Theory 20E26, 20E08 (Primary) 20E06 (Secondary) |
| url | https://arxiv.org/abs/2405.09736 |