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| Natura: | Preprint |
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2021
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| Accesso online: | https://arxiv.org/abs/2110.09822 |
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| _version_ | 1866911438296055808 |
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| author | Genevois, Anthony Tessera, Romain |
| author_facet | Genevois, Anthony Tessera, Romain |
| contents | Given a morphism $φ: G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $φ$ is a sufficiently large subgroup, then $\mathrm{ker}(φ)$ contains a non-abelian free subgroup and $φ$ factors through an acylindrically hyperbolic quotient of $G$. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier-Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_09822 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A note on morphisms to wreath products Genevois, Anthony Tessera, Romain Group Theory 20F65, 20F67 Given a morphism $φ: G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $φ$ is a sufficiently large subgroup, then $\mathrm{ker}(φ)$ contains a non-abelian free subgroup and $φ$ factors through an acylindrically hyperbolic quotient of $G$. As direct applications, we classify the finitely presented subgroups in $A \wr B$ up to isomorphism and we deduce that a group having a wreath product $(\text{non-trivial}) \wr (\text{infinite})$ as a quotient must be SQ-universal (extending theorems of Baumslag and Cornulier-Kar). Finally, we exploit our theorem in order to describe the structure of the automorphism groups of several families of wreath products, highlighting an interesting connection with the Kaplansky conjecture on units in group rings. |
| title | A note on morphisms to wreath products |
| topic | Group Theory 20F65, 20F67 |
| url | https://arxiv.org/abs/2110.09822 |