Newelski's Conjecture for $o$-Minimal and $p$-Adic Groups
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| Formato: | Preprint |
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2026
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| _version_ | 1866914301038559232 |
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| author | Yao, Ningyuan Zhang, Zhentao |
| author_facet | Yao, Ningyuan Zhang, Zhentao |
| contents | 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 $M_0 = \mathbb{Q}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Newelski's Conjecture for $o$-Minimal and $p$-Adic Groups Yao, Ningyuan Zhang, Zhentao Logic 03C65 03C98 37B05 11E95 03C64 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 $M_0 = \mathbb{Q}_p$. |
| title | Newelski's Conjecture for $o$-Minimal and $p$-Adic Groups |
| topic | Logic 03C65 03C98 37B05 11E95 03C64 |
| url | https://arxiv.org/abs/2602.01810 |