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• W4379986402 abstract "Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{mathbb {Q}}_p}$ , with “definable f -generics” in the sense of an ambient saturated elementary extension of ${{mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f -generic” or $dfg$ group we mean a definable group in a saturated model with a global f -generic type which is definable over a small model. In the present context the group is definable over ${{mathbb {Q}}_p}$ , and the small model will be ${{mathbb {Q}}_p}$ itself. The notion of a $mathrm {dfg}$ group is dual, or rather opposite to that of an $operatorname {mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o -minimal groups in the p -adic context. In the current paper our group will be definable over ${{mathbb {Q}}_p}$ in an ambient saturated elementary extension $mathbb {K}$ of ${{mathbb {Q}}_p}$ , so as to make sense of the notions of f -generic type, etc. In this paper we will show that every definable f -generic group definable in ${{mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{mathbb {Q}}_p}$ . This is analogous to the o -minimal context, where every connected torsion-free group definable in $mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f -generic types coincide with almost periodic types in the p -adic case." @default.
• W4379986402 created "2023-06-10" @default.
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• W4379986402 date "2023-06-09" @default.
• W4379986402 modified "2023-09-24" @default.
• W4379986402 title "ON GROUPS WITH DEFINABLE <i>F</i>-GENERICS DEFINABLE IN <i>P</i>-ADICALLY CLOSED FIELDS" @default.
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• W4379986402 doi "https://doi.org/10.1017/jsl.2023.37" @default.
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