On definable $f$-generic groups and minimal flows in $p$-adically closed fields

Abstract

Let $X$ be a definable group definable over a small model $M_0$. Recall that a global type $p$ on $X$ is definable $f$-generic over $M_0$ if every left translate of $p$ is definable over $M_0$. We call $p$ strongly $f$-generic over $M_0$ if every left translate of $p$ does not fork over $M_0$. Let $H$ be a group definable over the field ${\mathbb{Q}}_p$ of $p$-adic numbers admitting global definable $f$-generic types over ${\mathbb{Q}}_p$. We show that $H$ has unboundedly many global weakly generic types iff there is a global type $r$ on $H$ which is strongly $f$-generic over ${\mathbb{Q}}_p$ and a ${\mathbb{Q}}_p$-definable function $\theta$ such that $\theta (r)$ is finitely satisfiable in ${\mathbb{Q}}_p$. Recall that the $\mu$-type $\mu (x)$ on $H$ is the partial type consisting of the formulas over ${\mathbb{Q}}_p$ which define open neighborhoods of the identity of $H$. We show that every global weakly generic type $r$ on $H$ is $\mu$-invariant: For any $\epsilon \models \mu$ and $a\models r$, we have $\epsilon \cdot a\models r$. Let $G$ be groups definable over ${\mathbb{Q}}_p$ such that $H$ is a normal subgroup of $G$ and $G/H$ is a definably compact group. Then we show that the weakly generic types on $G$ coincide with almost periodic types $G$ iff $G$ has boundedly many global weakly generic types.