Semisimple groups interpretable in various valued fields
We study infinite groups interpretable in power bounded T-convex, V-minimal or p-adically closed fields. We show that if G is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a gro...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509425100844/type/journal_article |
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| author | Yatir Halevi Assaf Hasson Ya'acov Peterzil |
| author_facet | Yatir Halevi Assaf Hasson Ya'acov Peterzil |
| author_sort | Yatir Halevi |
| collection | DOAJ |
| description | We study infinite groups interpretable in power bounded T-convex, V-minimal or p-adically closed fields. We show that if G is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group
$G_1\times G_2$
, where
$G_1$
is a K-linear group and
$G_2$
is a
$\mathbf {k}$
-linear group. The analysis is carried out by studying the interaction of G with four distinguished sorts: the valued field K, the residue field
$\mathbf {k}$
, the value group
$\Gamma $
, and the closed
$0$
-balls
$K/\mathcal {O}$
. |
| format | Article |
| id | doaj-art-d10f2d56d47c4ffa8ed4aba2153a3903 |
| institution | DOAJ |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-d10f2d56d47c4ffa8ed4aba2153a39032025-08-20T02:47:43ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2025.10084Semisimple groups interpretable in various valued fieldsYatir Halevi0https://orcid.org/0000-0001-8423-7504Assaf Hasson1Ya'acov Peterzil2Faculty of Natural Sciences, Department of Mathematics, https://ror.org/02f009v59 University of Haifa , Haifa, IsraelDepartment of Mathematics, https://ror.org/05tkyf982 Ben Gurion University of the Negev , Be’er-Sheva, Israel; E-mail:Faculty of Natural Sciences, Department of Mathematics, https://ror.org/02f009v59 University of Haifa , Haifa, Israel; E-mail:We study infinite groups interpretable in power bounded T-convex, V-minimal or p-adically closed fields. We show that if G is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$ , where $G_1$ is a K-linear group and $G_2$ is a $\mathbf {k}$ -linear group. The analysis is carried out by studying the interaction of G with four distinguished sorts: the valued field K, the residue field $\mathbf {k}$ , the value group $\Gamma $ , and the closed $0$ -balls $K/\mathcal {O}$ .https://www.cambridge.org/core/product/identifier/S2050509425100844/type/journal_article12L1203C60 |
| spellingShingle | Yatir Halevi Assaf Hasson Ya'acov Peterzil Semisimple groups interpretable in various valued fields Forum of Mathematics, Sigma 12L12 03C60 |
| title | Semisimple groups interpretable in various valued fields |
| title_full | Semisimple groups interpretable in various valued fields |
| title_fullStr | Semisimple groups interpretable in various valued fields |
| title_full_unstemmed | Semisimple groups interpretable in various valued fields |
| title_short | Semisimple groups interpretable in various valued fields |
| title_sort | semisimple groups interpretable in various valued fields |
| topic | 12L12 03C60 |
| url | https://www.cambridge.org/core/product/identifier/S2050509425100844/type/journal_article |
| work_keys_str_mv | AT yatirhalevi semisimplegroupsinterpretableinvariousvaluedfields AT assafhasson semisimplegroupsinterpretableinvariousvaluedfields AT yaacovpeterzil semisimplegroupsinterpretableinvariousvaluedfields |