Newelski's Conjecture for $o$-Minimal and $p$-Adic Groups

Abstract

Let $ M_0 $ denote either the field structure $ \mathbb{Q}_p $ of $ p $-adic numbers, or an $o$-minimal expansion of the field structure $ \mathbb{R} $ of real numbers. We investigate the minimal flows and Ellis groups of definable groups over $ M_0 $ from the perspective of definable topological dynamics. This paper builds on the research initiated in \cite{BY-APAL} and generalizes the main results thereof in two key ways: First, we extend the scope of these results from reductive algebraic groups to arbitrary definable groups. Second, we generalize the approach from $ p $-adically closed fields to $o$-minimal expansions of real closed fields. Let $G$ be a definable group over $M_0$, and let $B$ be a definably amenable component (see Definition \ref{def-DAC}) of $G$. In a certain sense, $B$ can be regarded as a ``maximal definably amenable subgroup'' of $G$ (see Fact \ref{fact-max-DA-subgroup}). The main conclusion of this paper is as follows: For any $M \succ M_0$, the Ellis group of the universal definable flow of $G$ over $M$ is isomorphic to that of $B$ over $M$. In particular, the Ellis groups of the universal definable flow of $G$ are model-independent, as is the case for $B$ (see \cite{CS-Definably-Amenable-NIP-Groups}). As a consequence, we conclude that Newelski's Conjecture holds if and only if $G$ is definably amenable when $M_0 = \mathbb{Q}_p$.