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

TL;DR

This work advances the understanding of definable topological dynamics for groups in two taming contexts: $p$-adic fields and $o$-minimal expansions of real closed fields. It shows that for any definable group $G$ over $M_0$, the Ellis group of the universal definable flow over any $M\succ M_0$ is isomorphic to the Ellis group of a definably amenable component $B$, yielding model-independence of the Ellis group and enabling a reduction of Newelski's Conjecture to $B$. A key contribution is the Amenable-Semisimple Decomposition, which yields a short exact sequence with a definably amenable kernel and a semisimple quotient, and a detailed construction of the Ellis group via the dfg component and infinitesimals. The paper proves that, in the $\mathbb{Q}_p$ setting, Newelski's Conjecture for $G$ is equivalent to $G$ being definably amenable, and it provides explicit descriptions of the Ellis groups in terms of $B/B^{00}$ and related quotients. Collectively, these results unify the computation of Ellis groups across $o$-minimal and $p$-adic settings and clarify when Newelski's Conjecture holds, highlighting the central role of the definably amenable component and the dfg/infinitesimal structure.

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 $M_0 = \mathbb{Q}_p$.

Theorems & Definitions (92)

• Theorem 1: CS-Definably-Amenable-NIP-Groups Theorem 5.6
• Theorem 2: = Theorem \ref{['theorem-rG*eb']}
• Theorem 3: = Theorem \ref{["thm-Newelski's Conjecture=DA"]}
• Remark 2.3.6
• Corollary 2.3.11
• Remark 2.4.5
• Theorem 4
• Remark 2.4.6
• Theorem 5: PYZ-$p$-adic-groups
• Lemma 2.5.3