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 dfg group is dual, or rather opposite to that of an 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 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 (Lemma 3.4, \cite{COS}). 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 \cite{P-Y} of whether $f$-generic types coincide with almost periodic types in the $p$-adic case.