Stabilizers, groups with f-generics in NTP2 and PRC fields

Authors

Samaria Montenegro, Alf Onshuus, Pierre Simon

Type: Preprint

Publication Date: 2016-01-01

Citations: 7

DOI: https://doi.org/10.48550/arxiv.1610.03150

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Abstract

In this paper we develop three different subjects. We study and prove alternative versions of Hrushovski's "Stabilizer Theorem", we generalize part of the basic theory of definably amenable NIP groups to NTP2 theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded PRC fields.

Locations

arXiv (Cornell University)
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Groups with f-generics in NTP2 and PRC fields

2018-01-01

Samaria Montenegro Guzmán , Alf Onshuus Niño , Simon Pierre Sigué

We study groups with f-generic types definable in bounded PRC fields. Along the way, we generalize part of the basic theory of definably amenable NIP groups to NTP$_2$ theories and … We study groups with f-generic types definable in bounded PRC fields. Along the way, we generalize part of the basic theory of definably amenable NIP groups to NTP$_2$ theories and prove variations on Hrushovski's stabilizer theorem.

STABILIZERS, GROUPS WITH -GENERICS, AND PRC FIELDS

2018-05-10

Samaria Montenegro , Alf Onshuus , Pierre Simon

In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups … In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.

On f-generic types in NIP groups

2023-01-01

Atticus Stonestrom

Let $G$ be a group definable in an NIP theory. We prove that, if $G$ admits a global f-generic type, then $G$ is definably amenable, answering a question of Chernikov … Let $G$ be a group definable in an NIP theory. We prove that, if $G$ admits a global f-generic type, then $G$ is definably amenable, answering a question of Chernikov and Simon. As an application, we show that every dp-minimal group is definably amenable, answering a question of Kaplan, Levi, and Simon.

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NIP and definably amenable groups

2018-04-06

Pierre Simon

This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly \[6], \[7] and \[4]. This subject has two origins, the … This text is an introduction to de finably amenable NIP groups. It is based on a number of papers, mainly \[6], \[7] and \[4]. This subject has two origins, the fi rst one is the theory of stable groups and in particular generic types, which were fi rst defi ned by Poizat (see \[12]) and have since played a central role throughout stability theory. Later, part of the theory was generalized to groups in simple theories, where generic types are de ned as types, none of whose translates forks over the empty set.

On Amalgamation in NTP2 Theories and Generically Simple Generics

2020-02-22

Pierre Simon

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. … We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

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On groups and fields interpretable in $\mathrm{NTP_2}$ fields

2024-09-16

Paul Wang

This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic … This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.

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Groups in NTP2

2015-01-01

Nadja Hempel , Alf Onshuus

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory. We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.

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Groups in NTP2

2015-10-05

Nadja Hempel , Alf Onshuus

Definably amenable NIP groups

2015-02-15

Artem Chernikov , Pierre Simon

We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, … We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

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Definably amenable NIP groups

2015-01-01

Artem Chernikov , Pierre Simon

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Definably amenable NIP groups

2017-12-27

Artem Chernikov , Pierre Simon

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, … We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

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

2019-01-01

Anand Pillay , Ningyuan Yao

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 … 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.

Definable $f$-Generic Groups over $p$-Adic Numbers

2019-11-04

Anand Pillay , Ningyuan Yao

The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group … The aim of this paper is to develop the theory for \emph{definable $f$-generic} groups in the $p$-adic field within the framework of topological dynamics, here the definable means a group admits a global f-generic type which is over a small submodel. This definable is a dual concept to finitely satisfiable generic, and a useful tool to describe the analogue of torsion free o-minimal groups in the $p$-adic context. In this paper we will show that every $f$-generic group in $\Q$ is eventually isomorphic to a finite index subgroup of a trigonalizable algebraic group over $\Q$. This is analogous to the $o$-minimal context, where every connected torsion free group in $\R$ is isomorphic to a trigonalizable algebraic group (Lemma 3.4, \cite{COS}). We will also show that every open $f$-generic subgroup of a $f$-generic group has finite index, and every $f$-generic type of a $f$-generic group is almost periodic, which gives a positive answer on the problem raised in \cite{P-Y} of whether $f$-generic types coincide with almost periodic types in the $p$-adic case.

Topological dynamics and NIP fields

2021-06-02

Grzegorz Jagiella

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Topological dynamics and NIP fields

2020-01-01

Grzegorz Jagiella

We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field $K$. We show that the Ellis group of the universal definable flow of $\mathrm{SL}_2(K)$ is … We study definable topological dynamics of some algebraic group actions over an arbitrary NIP field $K$. We show that the Ellis group of the universal definable flow of $\mathrm{SL}_2(K)$ is non-trivial if the multiplicative group of $K$ is not type-definably connected, providing a way to find multiple counterexamples to the Ellis group conjecture, particularly in the case of dp-minimal fields. We also study some structure theory of algebraic groups over $K$ with definable f-generics.

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Topological dynamics and NIP fields

2020-10-27

Grzegorz Jagiella

Groups and fields with NTP2

2012-01-01

Artem Chernikov , Itay Kaplan , Pierre Simon

NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain … NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in NTP2 structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP2 field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight), and show that every strongly dependent valued field is Kaplansky.

Groups and fields with NTP2

2012-12-26

Artem Chernikov , Itay Kaplan , Pierre Simon

The Projective Height Zero Conjecture

2017-12-22

Gunter Malle , Gabriel Navarro

We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for … We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.

The Projective Height Zero Conjecture

2017-01-01

Gunter Malle , Gabriel Navarro

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Artin-Schreier extensions and combinatorial complexity in henselian valued fields

2024-09-03

Blaise Boissonneau

Abstract We give explicit formulas witnessing IP, IP $_{\!n}$ , or TP2 in fields with Artin–Schreier extensions. We use them to control p -extensions of mixed characteristic henselian valued fields, … Abstract We give explicit formulas witnessing IP, IP $_{\!n}$ , or TP2 in fields with Artin–Schreier extensions. We use them to control p -extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP $_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP $_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

On piecewise hyperdefinable groups

2022-12-16

A. Rodriguez Fanlo

The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise … The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.

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On compactifications and product‐free sets

2019-07-24

Daniel Palacín

A subset of a group is said to be product-free if it does not contain three elements satisfying the equation $xy=z$. We give a negative answer to a question of … A subset of a group is said to be product-free if it does not contain three elements satisfying the equation $xy=z$. We give a negative answer to a question of Babai and S\'os on the existence of large product-free sets by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product-free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.

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A note on groups definable in the p-adic field

2018-01-01

Anand Pillay , Ningyuan Yao

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 … 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.

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One dimensional commutative groups definable in algebraically closed valued fields and in the pseudo-local fields.

2021-12-01

Juan Pablo Acosta

We give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by … We give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by a finite subgroup.

Auxiliary Beam Pair Enabled AoD and AoA Estimation in Closed-Loop Large-Scale mmWave MIMO Systems

2016-01-01

Dalin Zhu , Junil Choi , Robert W. Heath

Channel estimation is of critical importance in millimeter-wave (mmWave) multiple-input multiple-output (MIMO) systems. Due to the use of large antenna arrays, low-complexity mmWave specific channel estimation algorithms are required. In … Channel estimation is of critical importance in millimeter-wave (mmWave) multiple-input multiple-output (MIMO) systems. Due to the use of large antenna arrays, low-complexity mmWave specific channel estimation algorithms are required. In this paper, an auxiliary beam pair design is proposed to provide high-resolution estimates of the channel's angle-of-departure (AoD) and angle-of-arrival (AoA) for mmWave MIMO systems. By performing an amplitude comparison with respect to each auxiliary beam pair, a set of ratio measures that characterize the channel's AoD and AoA are obtained by the receiver. Either the best ratio measure or the estimated AoD is quantized and fed back to the transmitter via a feedback channel. The proposed technique can be incorporated into control channel design to minimize initial access delay. Though the design principles are derived assuming a high-power regime, evaluation under more realistic assumption shows that by employing the proposed method, good angle estimation performance is achieved under various signal-to-noise ratio levels and channel conditions.

Some NIP-like phenomena in NTP$_{2}$

2017-01-01

Itay Kaplan , Pierre Simon

We introduce the notion of an NTP$_{2}$-smooth measure and prove that they exist assuming NTP$_{2}$. Using this, we propose a notion of distality in NTP$_{2}$ that unfortunately does not intersect … We introduce the notion of an NTP$_{2}$-smooth measure and prove that they exist assuming NTP$_{2}$. Using this, we propose a notion of distality in NTP$_{2}$ that unfortunately does not intersect simple theories trivially. We then prove a finite alternation theorem for a subclass of NTP$_{2}$ that contains resilient theories. In the last section we prove that under NIP, any type over a model of singular size is finitely satisfiable in a smaller model, and ask if a parallel result (with non-forking replacing finite satisfiability) holds in NTP$_{2}$.

A Guide to NIP Theories

2015-07-05

Pierre Simon

The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to … The study of NIP theories has received much attention from model theorists in the last decade, fuelled by applications to o-minimal structures and valued fields. This book, the first to be written on NIP theories, is an introduction to the subject that will appeal to anyone interested in model theory: graduate students and researchers in the field, as well as those in nearby areas such as combinatorics and algebraic geometry. Without dwelling on any one particular topic, it covers all of the basic notions and gives the reader the tools needed to pursue research in this area. An effort has been made in each chapter to give a concise and elegant path to the main results and to stress the most useful ideas. Particular emphasis is put on honest definitions, handling of indiscernible sequences and measures. The relevant material from other fields of mathematics is made accessible to the logician.

TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01

A. J. Wilkie

By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).

Stable group theory and approximate subgroups

2011-06-15

Ehud Hrushovski

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math … We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

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On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.

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