ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

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.