Embeddings into highly transitive and mixed identity free groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916947229147136 |
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| author | Hyde, James Lodha, Yash |
| author_facet | Hyde, James Lodha, Yash |
| contents | Given a countable group $G$, we develop a method to construct an overgroup $H$ that is finitely generated, highly transitive and mixed identity free. Our construction can be controlled to ensure that some fundamental group theoretic properties of $G$ are inherited by $H$, such as amenability or the property of not containing a nonabelian free group. The former provides a strong solution to a question of Hull and Osin, and the latter provides the first examples of nonamenable groups without free subgroups that are highly transitive and mixed identity free. Our examples also have a nontrivial amenable radical, answering a question of Arzhantseva. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Embeddings into highly transitive and mixed identity free groups Hyde, James Lodha, Yash Group Theory Given a countable group $G$, we develop a method to construct an overgroup $H$ that is finitely generated, highly transitive and mixed identity free. Our construction can be controlled to ensure that some fundamental group theoretic properties of $G$ are inherited by $H$, such as amenability or the property of not containing a nonabelian free group. The former provides a strong solution to a question of Hull and Osin, and the latter provides the first examples of nonamenable groups without free subgroups that are highly transitive and mixed identity free. Our examples also have a nontrivial amenable radical, answering a question of Arzhantseva. |
| title | Embeddings into highly transitive and mixed identity free groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2509.09788 |