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

Abstract

Let $M_0$ denote either the field of $p$-adic numbers ${\mathbb{Q}}_p$ or an $o$-minimal expansion of the real field $\mathbb{R}$. We study the minimal flows and Ellis groups of groups definable over $M_0$ from the viewpoint of definable topological dynamics. The computation of Ellis groups, together with the closely related Newelski conjecture, has so far been available only under restrictive hypotheses: in the $p$-adic setting, the results of \cite{BY-APAL} apply to reductive algebraic groups, while in the $o$-minimal setting the known computations rely on a compact-torsion-free (Iwasawa) decomposition. In this paper we remove both restrictions. We show that every definable group $G$ has a definably amenable radical $D$, with quotient map $\pi :G\to A$ onto a centreless semisimple group $A$. In $A$, a definably amenable component $V$ is selected: a maximal definably amenable subgroup containing a maximal definably solvable subgroup. Its preimage $B={\pi }^{-1}(V)$ is a definably amenable component of $G$, contains $D$, and is the carrier for the computation of the Ellis group. Our main result is that, for every $M\succ M_0$, the Ellis group of the universal definable flow of $G$ over $M$ is isomorphic to that of $B$ over $M$, and hence to $B/B^{00}$. In particular, the Ellis groups of $G$ are model-independent. As a consequence, when $G$ is definable over a $p$-adically closed field, Newelski's conjecture holds for $G$ if and only if $G$ is definably amenable. The amenable direction is the known result of \cite{CS-Definably-Amenable-NIP-Groups}, while the converse is new.