arXiv subjects

Ningyuan Yao

Publications and source records attributed to Ningyuan Yao.

17 recordsLinked to original sources

arXiv2026

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

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 $π:G\to A$ 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 $B=π^{-1}(V)$ 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.

math.LO↗

arXiv2026

On groups definable in $p$-adically closed fields

This paper is about the $dfg$/$fsg$ decomposition for groups $G$ definable in $p$-adically closed fields. It is proved that for $G$ definably amenable, $G$ has a definable normal $dfg$ subgroup $H$ such that the quotient $G/H$ is a definable $fsg$ group. The result was known for groups definable in $o$-minimal expansions of real closed fields (see \cite{C-P-o-mini}). We also give a version for arbitrary (not necessarily definably amenable) groups $G$ definable in $p$-adically closed fields: there is a definable $dfg$ subgroup $H$ of $G$ such that the homogeneous space $G/H$ is definable and definably compact. (In the $o$-minimal case this is Fact 3.25 of \cite{Peterzil-Starchenko-mutypes}). Finally, we also isolate the definably amenable part of $G$, which we call the definably amenable component. Note that $dfg$ stands for ``has a definable $f$-generic type", and $fsg$ for ``has finitely satisfiable generics", which will be discussed together with various equivalences. We will need to understand something about groups of the form $G(k)$ where $k$ is a $p$-adically closed field and $G$ a semisimple algebraic group over $k$, and as part of the analysis we will prove the Kneser-Tits conjecture over $p$-adically closed fields.

math.LO↗

arXiv2024

One-dimensional subgroups and connected components in non-abelian $p$-adic definable groups

We generalize two of our previous results on abelian definable groups in $p$-adically closed fields to the non-abelian case. First, we show that if $G$ is a definable group that is not definably compact, then $G$ has a one-dimensional definable subgroup which is not definably compact. This is a $p$-adic analogue of the Peterzil-Steinhorn theorem for o-minimal theories. Second, we show that if $G$ is a group definable over the standard model $\mathbb{Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb{Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when $G$ is a definable subgroup of a linear algebraic group, over any model.

math.LO↗

arXiv2023

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

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 $θ$ such that $θ(r)$ is finitely satisfiable in ${\mathbb Q}_p$. Recall that the $μ$-type $μ(x)$ 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 $μ$-invariant: For any $ε\models μ$ and $a\models r$, we have $ε\cdot a\models r$. 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.

math.LO↗

arXiv2023

Generic stability of linear algebraic groups over $\mathbb{C}[[t]]$

Let $K$ be a henselian valued field with ${\cal O}_K$ its valuation ring, $Γ$ its value group, and $\boldsymbol{k}$ its residue field. We study the definable subsets of ${\cal O}_K$ and algebraic groups definable over ${\cal O}_K$ in the case where $\boldsymbol{k}$ is algebraically closed and $Γ$ is a $\mathbb Z$-group. We first describe the definable subsets of ${\cal O}_K$, showing that every definable subset of ${\cal O}_K$ is either res-finite or res-cofinite (see Definition \ref{def-res-finite-cofinite}). Applying this result, we show that $\mathrm{GL}(n,{\cal O}_K)$ (the invertible $n$ by $n$ matrices over ${\cal O}_K$) are generically stable for each $n$, generalizing Y. Halevi's result, where $K$ is an algebraically closed valued field \cite{Y.Halevi}.

math.LO↗

arXiv2023

Open subgroups of $p$-adic algebraic groups

In \cite{Pillay} and more formally in \cite{Onshuus-Pillay} it was asked whether open subgroups of $p$-adic algebraic groups are ($p$-adic) semialgebraic, equivalently, definable in the structure $(\mathbb Q_{p}, +, \times)$. We give a positive answer in the commutative case. Together with results of \cite{Prasad} this leads to a positive answer for reductive algebraic groups.

math.GR↗

arXiv2023

On minimal flows of commutative $p$-adic groups

We study the definable topological dynamics $(G,S_G(M))$ of a definable group acting on its type space, where $M$ is a structure and $G$ is a group definable in $M$. In \cite{Newelski-I}, Newelski raised a question of whether weakly generic types coincide with almost periodic types in definable topological dynamics. In \cite{YZ-Sta}, we introduced the notion of stationarity, showing the answer is positive when $G$ is a stationary definably amenable group definable over the field of $p$-adic numbers or an $o$-minimal expansion of real closed field. In this paper, we continue with the work of \cite{YZ-Sta}, focusing on the case where $G$ is a commutative group definable over the field of $p$-adic numbers, and showing that weakly generic types coincide with almost periodic types if and only if either $G$ has definable $f$-generics or $G$ is stationary.

math.LO↗

arXiv2023

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable groups, and o-minimality" to a more general context.

math.LO↗

arXiv2023

On groups with definable $f$-generics definable in $p$-adically closed fields

The aim of this paper is to develop the theory of groups definable in the $p$-adic field ${\mathbb Q}_p$, with ``definable $f$-generics" in the sense of an ambient saturated elementary extension of ${\mathbb Q}_p$. We call such groups definable $f$-generic groups. So, by a ``definable f-generic'' or dfg group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${\mathbb Q}_p$, and the small model will be ${\mathbb Q}_p$ itself. The notion of a dfg group is dual, or rather opposite to that of an fsg group (group with ``finitely satisfiable generics") and is a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context. In the current paper our group will be definable over ${\mathbb Q}_p$ in an ambient saturated elementary extension $\mathbb K$ of ${\mathbb Q}_p$, so as to make sense of the notions of $f$-generic etc. In this paper we will show that every definable $f$-generic group definable in ${\mathbb Q}_p$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${\mathbb Q}_p$. This is analogous to the $o$-minimal context, where every connected torsion free group definable in $\mathbb R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open definable $f$-generic subgroup of a definable $f$-generic group has finite index, and every $f$-generic type of a definable $f$-generic group is almost periodic, which gives a positive answer to the problem raised in \cite{P-Y} of whether $f$-generic types coincide with almost periodic types in the $p$-adic case.

math.LO↗

arXiv2022

Abelian groups definable in $p$-adically closed fields

Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model $\mathbb{Q}_p$, we show that $G^0 = G^{00}$, and that $G$ is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb{Q}_p$.

math.LO↗

arXiv2022

On non-compact $p$-adic definable groups

Peterzil and Steinhorn proved that if a group $G$ definable in an $o$-minimal structure is not definably compact, then $G$ contains a definable torsion-free subgroup of dimension one. We prove here a $p$-adic analogue of the Peterzil-Steinhorn theorem, in the special case of abelian groups. Let $G$ be an abelian group definable in a $p$-adically closed field $M$. If $G$ is not definably compact then there is a definable subgroup $H$ of dimension one which is not definably compact. In a future paper we will generalize this to non-abelian $G$.

math.LO↗

arXiv2021

Definably Topological Dynamics of $p$-Adic Algebraic Groups

We study the $p$-adic algebraic groups $G$ from the definable topological-dynamical point of view. We consider the case that $M$ is an arbitrary $p$-adic closed field and $G$ an algebraic group over ${\mathbb Q}_p$ admitting an Iwasawa decompostion $G=KB$, where $K$ is open and definably compact over ${\mathbb Q}_p$, and $B$ is a borel subgroup of $G$ over ${\mathbb Q}_p$. Our main result is an explicit description of the minimal subflow and Ellis Group of the universal definable $G(M)$-flow $S_G(M^{\text{ext}})$. We prove that the Ellis group of $S_G(M^{\text{ext}})$ is isomorphic to the Ellis group of $S_B(M^{\text{ext}})$, which is $B/B^0$. As applications, we conclude that the Ellis groups corresponding to $\text{GL}(n,M)$ and $\text{SL}(n,M)$ are isomorphic to $(\hat {\mathbb Z} \times {\mathbb Z}_p^*)^n$ and $(\hat {\mathbb Z} \times {\mathbb Z}_p^*)^{n-1}$ respectively, generalizing the main result of Penazzi, Pillay, and Yao in Some model theory and topological dynamics of $p$-adic algebraic groups, Fundamenta Mathematicae, 247 (2019), pp. 191--216.

math.LO↗

arXiv2020

On Dimensions, Standard Part Maps, and $p$-Adically Closed Fields

The aim of this paper is to study the dimensions and standard part maps between the field of $p$-adic numbers ${{\mathbb Q}_p}$ and its elementary extension $K$ in the language of rings $L_r$. We show that for any $K$-definable set $X\subseteq K^m$, $\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(X\cap {{\mathbb Q}_p}^m)$. Let $V\subseteq K$ be convex hull of $K$ over ${{\mathbb Q}_p}$, and $\text{\st}: V\rightarrow {{\mathbb Q}_p}$ be the standard part map. We show that for any $K$-definable function $f:K^m\rightarrow K$, there is definable subset $D\subseteq{{\mathbb Q}_p}^m$ such that ${{\mathbb Q}_p}^m\backslash D$ has no interior, and for all $x\in D$, either $f(x)\in V$ and $\text{st}(f(\text{st}^{-1}(x)))$ is constant, or $f(\text{st}^{-1}(x))\cap V=\emptyset$. We also prove that $\text{dim}_K(X)\geq \text{dim}_{{\mathbb Q}_p}(\text{st}(X\cap V^m))$ for every definable $X\subseteq K^m$.

math.LO↗

arXiv2019

Some model theory and topological dynamics of p-adic algebraic groups

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields. We consider the additive and multiplicative groups of Q_p and Z_p, the group of upper triangular invertible 2\times 2 matrices, SL(2,Z_p), and, our main focus, SL(2,Q_p). In all cases we identify f-generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the ``Ellis group" of SL(2,Q_p)$ is the profinite completion of Z, yielding a counterexample to Newelski's conjecture with new features: G = G^{00} = G^{000} but the Ellis group is infinite. A final section deals with the action of SL(2,Q_p) on the type-space of the projective line over Q_p.

math.LO↗

arXiv2019

Definable Topological Dynamics for Trigonalizable Algebraic Groups over Qp

We study the flow (G(Qp); SG(Qp)) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f-generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We will give a description of of f-generic types of trigonalizable algebraic groups, and prove that every f-generic type is almost periodic.

math.LO↗

arXiv2018

A note on groups definable in the p-adic field

It is known that a group G definable in the field of p-adic numbers is definably locally isomorphic to the group of Q_p-points of a connected algebraic group H defined over Q_p. We show that if H is commutative then G is commutative-by-finite. It follows in particular that any one-dimensional group definable in Q_p is commutative-by-finite. The results extend to groups definable in p-adically closed fields.

math.LO↗

arXiv2015

On minimal flows. definably amenable groups, and o-minimality

We study definably amenable groups in NIP theories, and answer a question of Newelski (and also of Chernikov-Simon), by giving an example in the o-minimal context where weak generic types do not coincide with almost periodic types, equivalently where the union of the minimal subflows of suitable type spaces is not closed. We give other positive results in this o-minimal context.

math.LO↗