STABILIZERS, GROUPS WITH -GENERICS, AND PRC FIELDS

Authors

Samaria Montenegro, Alf Onshuus, Pierre Simon

Type: Article

Publication Date: 2018-05-10

Citations: 12

DOI: https://doi.org/10.1017/s147474801800021x

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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 $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.

Locations

Journal of the Institute of Mathematics of Jussieu
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Stabilizers, groups with f-generics in NTP2 and PRC fields

2016-01-01

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 NTP2 theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded PRC fields.

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.

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 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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Pseudo real closed fields, pseudo p-adically closed fields and NTP2

2016-10-01

Samaria Montenegro

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Generically stable and smooth measures in NIP theories

2012-12-13

Ehud Hrushovski , Anand Pillay , Pierre Simon

We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by … We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.

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Pseudo real closed field, pseudo p-adic closed fields and NTP2

2014-11-27

Samaria Montenegro

The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 … The main result of this paper is a positive answer to the Conjecture by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC-field, then Th(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then Th(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC(PpC)-field, then Th(M) is strong.

A class of finite resistant p-groups

2016-11-30

He Guo Liu , Yu Lei Wang

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

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

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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Topological dynamics and NIP fields

2021-06-02

Grzegorz Jagiella

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Residually finite PSP-groups

1999-01-01

A. Mohammadi Hassanabadi

Subgroups of the Group of Formal Power Series with the Big Powers Condition

2019-01-01

Alexander Brudnyi

We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely … We study the structure of discrete subgroups of the group $G[[r]]$ of complex formal power series under the operation of composition of series. In particular, we prove that every finitely generated fully residually free group is embeddable to $G[[r]]$.

Langlands-Rapoport Conjecture Over Function Fields

2016-01-01

Esmail Arasteh Rad , Urs Hartl

In this article we formulate and prove the analogue of the Langlands-Rapoport conjecture for the moduli stacks of global $G$-shtukas. Here $G$ is a parahoric Bruhat-Tits group scheme over a … In this article we formulate and prove the analogue of the Langlands-Rapoport conjecture for the moduli stacks of global $G$-shtukas. Here $G$ is a parahoric Bruhat-Tits group scheme over a smooth projective curve $C$ over a finite field $\mathbb{F}_q$.

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Pseudo T-closed fields

2023-01-01

Samaria Montenegro , Silvain Rideau

Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we … Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for studying them: the class of pseudo $T$-closed fields, where $T$ is an enriched theory of fields. These fields verify a "local-global" principle for the existence of points on varieties with respect to models of $T$. This approach also enables a good description of some fields equipped with multiple $V$-topologies, particularly pseudo algebraically closed fields with a finite number of valuations. One important result is a (model theoretic) classification result for bounded pseudo $T$-closed fields, in particular we show that under specific hypotheses on $T$, these fields are NTP$_2$ of finite burden.

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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.

ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

2023-06-09

Anand Pillay , Ningyuan Yao

Abstract 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 … Abstract 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 $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ 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 type, 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 [5, Lemma 3.4]. 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 [28] of whether f -generic types coincide with almost periodic types in the p -adic case.

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A note on groups definable in the p-adic field

2019-04-29

Anand Pillay , Ningyuan Yao

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THE GROUP CONFIGURATION THEOREM FOR GENERICALLY STABLE TYPES

2024-11-11

Paul Wang

Abstract We generalize Hrushovski’s group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically … Abstract We generalize Hrushovski’s group configuration theorem to the case where the type of the configuration is generically stable, without assuming tameness of the ambient theory. The properties of generically stable types, which we recall in the second section, enable us to adapt the proof known in the stable context.

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On groups interpretable in various valued fields

2024-06-19

Yatir Halevi , Assaf Hasson , Ya’acov Peterzil

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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ONE DIMENSIONAL GROUPS DEFINABLE IN THE p-ADIC NUMBERS

2021-02-15

Juan Pablo Acosta López

A complete list of one dimensional groups definable in the p-adic numbers is given, up to a finite index subroup and a quotient by a finite subgroup. A complete list of one dimensional groups definable in the p-adic numbers is given, up to a finite index subroup and a quotient by a finite subgroup.

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ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS

2023-07-18

Will Johnson , Ningyuan Yao

Abstract 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 … Abstract 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 $\mathbb {Q}_p$ .

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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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Complete type amalgamation for nonstandard finite groups

2024-04-30

Amador Martín-Pizarro , Daniel Palacín

We extend previous work on Hrushovski's stabilizer theorem and prove a measuretheoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types.This generalizes results of Gowers on products … We extend previous work on Hrushovski's stabilizer theorem and prove a measuretheoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types.This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups.We also obtain a model-theoretic proof of Roth's theorem on the existence of arithmetic progressions of length 3 for subsets of positive density in suitable definably amenable groups, such as countable amenable abelian groups without involutions and ultraproducts of finite abelian groups of odd order.

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Amenability, connected components, and definable actions

2021-12-08

Ehud Hrushovski , Krzysztof Krupiński , Anand Pillay

Abstract We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an … Abstract We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mrow> <mml:mspace /> <mml:mi>top</mml:mi> <mml:mspace /> </mml:mrow> <mml:mn>000</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group topologies on a given $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>00</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mspace /> <mml:mi>cl</mml:mi> <mml:mspace /> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>M</mml:mi> <mml:mn>000</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for any model M . Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G -invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∅</mml:mi> </mml:math> -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> in a suitable language (where $${{\mathbb {F}}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is the free group in 2-generators) for which the $$\bigvee $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⋁</mml:mo> </mml:math> -definable group $$H:=\langle X \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>⟨</mml:mo> <mml:mi>X</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

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Pseudo T-closed fields

2025-01-17

Samaria Montenegro , Silvain Rideau

Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework … Pseudo algebraically closed, pseudo real closed, and pseudo p-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately.We propose a unified framework for studying them: the class of pseudo T -closed fields, where T is an enriched theory of fields.These fields verify a "local-global" principle for the existence of points on varieties with respect to models of T. This approach also enables a good description of some fields equipped with multiple V -topologies, particularly pseudo algebraically closed fields with a finite number of valuations.One important result is a (model theoretic) classification result for bounded pseudo T -closed fields, in particular we show that under specific hypotheses on T, these fields are NTP 2 of finite burden.

Pseudo real closed fields

1981-01-01

Alexander Prestel

Weakly o-minimal structures and real closed fields

2000-04-13

Dugald Macpherson , David Marker , Charles Steinhorn

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures … A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

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The absolute galois group of a pseudo real closed field with finitely many orders

1985-01-01

Şerban A. Basarab

The algebraic nature of the elementary theory of PRC fields

1988-12-01

Moshe Jarden

Dependent first order theories, continued

2009-09-01

Saharon Shelah

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Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

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.

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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Paires de structures O-minimales

1998-06-01

Yerzhan Baisalov , Bruno Poizat

Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G … Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].

Forking and Dividing in NTP2 theories

2012-01-20

Artem Chernikov , Itay Kaplan

Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in … Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .

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Tame Topology and O-minimal Structures

1998-05-07

L. P. D. van den Dries

Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment … Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

Externally definable sets and dependent pairs

2012-05-12

Artem Chernikov , Pierre Simon

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Pseudo real closed fields, pseudo p-adically closed fields and NTP2

2016-10-01

Samaria Montenegro

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Definably compact groups definable in real closed fields. I

2017-03-24

Eliana Barriga

We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence … We study definably compact definably connected groups definable in a sufficiently saturated real closed field $R$. We introduce the notion of group-generic point for $\bigvee$-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group $G$ definable in $R$ there are a connected $R$-algebraic group $H$, a definable injective map $\phi$ from a generic definable neighborhood of the identity of $G$ into the group $H\left(R\right)$ of $R$-points of $H$ such that $\phi$ acts as a group homomorphism inside its domain. This result is used in [2] to prove that the o-minimal universal covering group of an abelian connected definably compact group definable in a sufficiently saturated real closed field $R$ is, up to locally definable isomorphisms, an open connected locally definable subgroup of the o-minimal universal covering group of the $R$-points of some $R$-algebraic group.

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.

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES

2014-03-01

Itaï Ben Yaacov , Artem Chernikov

Abstract We establish several results regarding dividing and forking in NTP 2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences … Abstract We establish several results regarding dividing and forking in NTP 2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP 2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP 2 . The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

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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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Forking and Dividing in Fields with Several Orderings and Valuations

2020-01-08

Will Johnson

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees … We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.

Model theoretic methods in the theory of topological fields.

1978-05-01

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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