Regular bi-interpretability and finite axiomatizability of Chevalley groups
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| Format: | Preprint |
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2023
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| _version_ | 1866909695105564672 |
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| author | Bunina, Elena Gvozdevsky, Pavel |
| author_facet | Bunina, Elena Gvozdevsky, Pavel |
| contents | In this paper we consider Chevalley groups over commutative rings with~$1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the system $G_2$ also $1/3$. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable (see Definition~2.2). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_01954 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regular bi-interpretability and finite axiomatizability of Chevalley groups Bunina, Elena Gvozdevsky, Pavel Group Theory Logic 20G35 (Primary) 20A15, 03C60 (Secondary) In this paper we consider Chevalley groups over commutative rings with~$1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the system $G_2$ also $1/3$. Under these assumptions we prove that the central quotients of Chevalley groups are regularly bi-interpretable with the corresponding rings, the class of all central quotients of Chevalley groups of a given type is elementarily definable and even finitely axiomatizable (see Definition~2.2). The same holds for adjoint Chevalley groups and for bondedly generated Chevalley groups. We also give an example of Chevalley group with infinite center, which is not bi-interpretable with the corresponding ring and is elementarily equivalent to a group that is not a Chevalley group itself. |
| title | Regular bi-interpretability and finite axiomatizability of Chevalley groups |
| topic | Group Theory Logic 20G35 (Primary) 20A15, 03C60 (Secondary) |
| url | https://arxiv.org/abs/2311.01954 |