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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2303.08165 |
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| _version_ | 1866913699860578304 |
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| author | Fisher, Sam P. Sánchez-Peralta, Pablo |
| author_facet | Fisher, Sam P. Sánchez-Peralta, Pablo |
| contents | Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is Hughes-free whenever $G$ is locally indicable. In particular, we prove that Kaplansky's Zero Divisor Conjecture holds for all group algebras of torsion-free $3$-manifold groups. The embedding is also used to confirm a conjecture of Kielak and Linton. Thanks to the work of Jaikin-Zapirain and Linton, another consequence of the embedding is that $kG$ is coherent whenever $G$ is a virtually compact special one-relator group.
If $G$ is a torsion-free one-relator group, let $\overline{kG}$ be the division ring containing $kG$ constructed by Lewin and Lewin. We prove that $\overline{kG}$ is Hughes-free whenever a Hughes-free $kG$-division ring exists. This is always the case when $k$ is of characteristic zero; in positive characteristic, our previous result implies that this happens when $G$ is virtually compact special. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08165 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Division rings for group algebras of virtually compact special groups and $3$-manifold groups Fisher, Sam P. Sánchez-Peralta, Pablo Group Theory 20F65, 12E15 Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is Hughes-free whenever $G$ is locally indicable. In particular, we prove that Kaplansky's Zero Divisor Conjecture holds for all group algebras of torsion-free $3$-manifold groups. The embedding is also used to confirm a conjecture of Kielak and Linton. Thanks to the work of Jaikin-Zapirain and Linton, another consequence of the embedding is that $kG$ is coherent whenever $G$ is a virtually compact special one-relator group. If $G$ is a torsion-free one-relator group, let $\overline{kG}$ be the division ring containing $kG$ constructed by Lewin and Lewin. We prove that $\overline{kG}$ is Hughes-free whenever a Hughes-free $kG$-division ring exists. This is always the case when $k$ is of characteristic zero; in positive characteristic, our previous result implies that this happens when $G$ is virtually compact special. |
| title | Division rings for group algebras of virtually compact special groups and $3$-manifold groups |
| topic | Group Theory 20F65, 12E15 |
| url | https://arxiv.org/abs/2303.08165 |