Complete Embeddings of Groups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912125445734400 |
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| author | Bridson, Martin R. Short, Hamish |
| author_facet | Bridson, Martin R. Short, Hamish |
| contents | Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of $G$. If $G$ has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08913 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complete Embeddings of Groups Bridson, Martin R. Short, Hamish Group Theory Geometric Topology 20F65, 20E08, 20F67, 57K32 Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of $G$. If $G$ has a finite presentation (respectively, a finite classifying space), then so does $G^*$. Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken. |
| title | Complete Embeddings of Groups |
| topic | Group Theory Geometric Topology 20F65, 20E08, 20F67, 57K32 |
| url | https://arxiv.org/abs/2312.08913 |