Commutators in groups definable in o-minimal structures

Authors

Elías Baro, Éric Jaligot, Margarita Otero

Type: Article

Publication Date: 2012-02-27

Citations: 12

DOI: https://doi.org/10.1090/s0002-9939-2012-11209-2

View Ask PDF

Abstract

We prove the definability and actually the finiteness of the commutator width of many commutator subgroups in groups definable in o-minimal structures. This applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Locations

Proceedings of the American Mathematical Society
View PDF
arXiv (Cornell University)
View PDF
CiteSeer X (The Pennsylvania State University)
View PDF

Ask a Question About This Paper

Summary

Commutators in groups definable in o-minimal structures

2010-01-01

E. Baro , E. Jaligot , M. Otero

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and … We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

Chat View PDF

Commutators in groups definable in o-minimal structures

2010-06-01

Elías Baro , Éric Jaligot , Margarita Otero

Chat View PDF

Groups definable in o-minimal structures: various properties and a diagram

2020-01-01

Annalisa Conversano

We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and … We present a diagram surveying equivalence or strict implication for properties of different nature (algebraic, model theoretic, topological, etc.) about groups definable in o-minimal structures. All results are well-known and an extensive bibliography is provided.

Chat View PDF

Commutator width in Chevalley groups

2012-06-11

Roozbeh Hazrat , A. V. Stepanov , N. A. Vavilov +1 more

The present paper is the [slightly expanded] text of our talk at the Conference Advances in Group Theory and Applications at Porto Cesareo in June 2011. Our main results assert … The present paper is the [slightly expanded] text of our talk at the Conference Advances in Group Theory and Applications at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

Chat View PDF

Commutator width in Chevalley groups

2012-01-01

Roozbeh Hazrat , A. V. Stepanov , N. A. Vavilov +1 more

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert … The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

Chat View PDF

Linear Groups Definable in o-Minimal Structures

2002-01-01

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Simple groups definable in O-minimal structures

2017-03-02

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

Chat View PDF

Simple groups definable in O-minimal structures

1998-01-01

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

Chat View PDF

Groups Definable in o-minimal Structures: Various Properties and a Diagram.

2021-01-01

Annalisa Conversano

Groups definable in linear O-minimal structures

2007-06-29

Pantelis E. Eleftheriou

A Jordan-Chevalley decomposition beyond algebraic groups

2022-01-01

Annalisa Conversano

A generalization of the Jordan-Chevalley decomposition of linear algebraic groups to definable groups in o-minimal structures is considered. As an application, we give a description of the commutator subgroup of … A generalization of the Jordan-Chevalley decomposition of linear algebraic groups to definable groups in o-minimal structures is considered. As an application, we give a description of the commutator subgroup of a definable group G, and show that when G has definable Levi subgroups, [G, G] is definable. Along the way, we identify crucial central subgroups, study disconnected groups, definable p-groups and 0-groups, abstractly compact subgroups, the Weyl group, and discuss a couple of other structure decompositions: 1) a compact-torsionfree decomposition and 2) a Levi decomposition. It turns out that definable p-groups are solvable, like finite p-groups, but they are not necessarily nilpotent.

Chat View PDF

On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

2011-11-09

Annalisa Conversano , Anand Pillay

We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed … We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.

Chat View PDF

On Levi subgroups and the Levi decomposition for groups definable in o-minimal structures

2011-01-01

Annalisa Conversano , Anand Pillay

Chat View PDF

Linearity of groups definable in o-minimal structures

2016-07-08

Olivier Frécon

On torsion-free groups in o-minimal structures

2005-10-01

Ya’acov Peterzil , Sergei Starchenko

We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably … We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .

Chat View PDF

Locally definable groups in o-minimal structures

2005-05-18

Mário J. Edmundo

Chat View PDF

On solvable groups and rings definable in o-minimal structures

2017-03-29

Mário J. Edmundo

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

One-dimensional groups definable in o-minimal structures

1994-10-01

Adam Strzeboński

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Corrigendum to “locally definable groups in o-minimal structures"

2008-01-01

Elías González , Mário J. Edmundo

Cartan subgroups and regular points of o‐minimal groups

2019-04-01

Elías Baro , Alessandro Berarducci , Margarita Otero

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M … Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field, we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup — a Cartan subgroup of G — and secondly that the set of regular points of G — a dense subset of G — is formed by points which belong to a unique Cartan subgroup of G.

Chat View PDF

The derived subgroup of linear and simply-connected o-minimal groups

2019-02-26

Elías Baro

Chat View PDF

A nonabelian Brunn–Minkowski inequality

2023-07-05

Yifan Jing , Chieu-Minh Tran , Ruixiang Zhang

Abstract Henstock and Macbeath asked in 1953 whether the Brunn–Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, … Abstract Henstock and Macbeath asked in 1953 whether the Brunn–Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

Chat View PDF

Groups definable in o-minimal structures: various properties and a diagram

2020-01-01

Annalisa Conversano

Chat View PDF

Linearity of groups definable in o-minimal structures

2016-07-08

Olivier Frécon

Cartan subgroups of groups definable in o-minimal structures

2013-11-28

Elías Baro , Éric Jaligot , Margarita Otero

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the … We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.

Chat View PDF

A nonabelian Brunn-Minkowski inequality

2021-01-01

Yifan Jing , Chieu-Minh Tran , Ruixiang Zhang

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and … Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions along the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.

Chat View PDF

Real Lie groups and o-minimality

2021-09-22

Annalisa Conversano , Alf Onshuus , Sacha Post

We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group … We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field. For any such group, we find a Lie-isomorphic group definable in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Subscript exp"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>exp</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {R}_{\exp }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which any Lie automorphism is definable.

Chat View PDF

A definability criterion for connected Lie groups

2022-03-22

Alf Onshuus , Sacha Post

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be … It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G $G$ is Lie isomorphic to a group definable in an o $o$ -minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G $G$ has its Levi subgroups with finite center, we prove that in order for G $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452).

Chat View PDF

SOLVABLE LIE GROUPS DEFINABLE IN O-MINIMAL THEORIES

2016-04-28

Annalisa Conversano , Alf Onshuus , Sergei Starchenko

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field. In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

Chat View PDF

Quasi groupes de Frobenius dimensionnels

2023-06-26

Samuel Zamour

Nous nous intéressons à une classe de groupes, les quasi groupes de Frobenius (avec involutions), dont la structure interne généralise celle des groupes classiques GA1(C), PGL2(C) et SO3(R) : un … Nous nous intéressons à une classe de groupes, les quasi groupes de Frobenius (avec involutions), dont la structure interne généralise celle des groupes classiques GA1(C), PGL2(C) et SO3(R) : un sous-groupe et ses conjugués, d'indice fini dans leur normalisateur et d'intersection mutuelle triviale, recouvrent "génériquement" le groupe ambiant.Dans la perspective de la théorie des modèles, nous travaillons avec l'hypothèse de l'existence d'une bonne notion dimension sur les ensembles définissables (il faut distinguer le cas o-minimal et le cas rangé).Nous accordons une attention particulière au cas rangé.En étudiant la géométrie d'incidence induite par les involutions, nous esquissons une classification des quasi groupes de Frobenius et nous déterminons ainsi sous quelles conditions des groupes classiques peuvent être identifiés dans un cadre dimensionnel.

Chat View PDF

A Jordan–Chevalley decomposition beyond algebraic groups

2025-06-01

Annalisa Conversano

Abstract We prove a decomposition of definable groups in o‐minimal structures generalizing the Jordan–Chevalley decomposition of linear algebraic groups. It follows that any definable linear group is a semidirect product … Abstract We prove a decomposition of definable groups in o‐minimal structures generalizing the Jordan–Chevalley decomposition of linear algebraic groups. It follows that any definable linear group is a semidirect product of its maximal normal definable torsion‐free subgroup and a definable subgroup , unique up to conjugacy, definably isomorphic to a semialgebraic group. Along the way, we establish two other fundamental decompositions of classical groups in arbitrary o‐minimal structures: (1) a Levi decomposition and (2) a key decomposition of disconnected groups, relying on a generalization of Frattini's argument to the o‐minimal setting. In o‐minimal structures, together with ‐groups, 0‐groups play a crucial role. We give a characterization of both classes and show that definable ‐groups are solvable, like finite ‐groups, but they are not necessarily nilpotent. Furthermore, we prove that definable ‐groups ( or prime) are definably generated by torsion elements and, in definably connected groups, 0‐Sylow subgroups coincide with ‐Sylow subgroups for each prime.

A survey on groups definable in o-minimal structures

2008-05-22

Margarita Otero

Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable … Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:

Chat View PDF

Lie groups and algebraic groups

1990-01-01

É. B. Vinberg , A. L. Onishchik , Dimitry Leites

On $ω_1$-categorical theories of abelian groups

1971-01-01

Angus Macintyre

Chat View PDF

Groups of Finite Morley Rank

1994-12-01

Alexandre Borovik , Ali Nesin

Abstract The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed … Abstract The book is devoted to the theory of groups of finite Morley rank. These groups arise in model theory and generalize the concept of algebraic groups over algebraically closed fields. The book contains almost all the known results in the subject. Trying to attract pure group theorists in the subject and to prepare the graduate student to start the research in the area, the authors adopted an algebraic and self evident point of view rather than a model theoretic one, and developed the theory from scratch. All the necessary model theoretical and group theoretical notions are explained in length. The book is full of exercises and examples and one of its chapters contains a discussion of open problems and a program for further research.

Definably simple groups in o-minimal structures

2000-02-24

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

Simple algebraic and semialgebraic groups over real closed fields

2000-06-13

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Lie-like decompositions of groups definable in o-minimal structures

2009-01-01

Annalisa Conversano

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study … There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

Chat View PDF

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

The Ore conjecture

2010-06-02

Martin W. Liebeck , E. A. O’Brien , Aner Shalev +1 more

The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of … The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Despite considerable effort, it remains open for various infinite families of simple groups. In this paper we develop new strategies, combining character theoretic methods with other ingredients, and use them to establish the conjecture. Liebeck acknowledges the support of a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Applications. O’Brien acknowledges the support of an LMS Visitor Grant, the Marsden Fund of New Zealand (grant UOA 0721), and the Mathematical Sciences Research Institute (Berkeley). Shalev acknowledges the support of an EPSRC Visiting Fellowship, an Israel Science Foundation Grant, and a Bi-National Science Foundation grant United States-Israel 2004-052. Tiep acknowledges the support of the NSF (grant DMS-0600967), and the Mathematical Sciences Research Institute (Berkeley).

Groups with the minimal condition on centralizers

1979-10-01

R. M. Bryant

On a result of Baer

1962-01-01

Maxwell Rosenlicht

3. J. M. G. Fell, The dual spaces of C*-algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403. 4. , C*-algebras with smooth dual, Illinois J. Math. 4 (1960), 221-230. 5. … 3. J. M. G. Fell, The dual spaces of C*-algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403. 4. , C*-algebras with smooth dual, Illinois J. Math. 4 (1960), 221-230. 5. , Weak containment and induced representations of groups, Canad. J. Math. (to appear). 6. J. G. Glimm, Type I C*-algebras, Ann. of Math. (2) 73 (1961), 572-612. 7. A. A. Kirillov, On unitary representations of nilpotent Lie groups, Doki. Akad. Nauk SSSR 130 (1959), 966-968. 8. G. W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. 55 (1952), 101-139. 9. , Borel structure in groups and their duals, Trans. Amer. Math. Soc. 85 (1957), 134-165. 10. , Unitary representations of group extensions. I, Acta Math. 99 (1958), 265-311. 11. Alex Rosenberg, The number of irreducible representations of simple rings with no minimal ideals, Amer. J. Math. 75 (1953), 523-530. 12. 0. Takenouchi, Sur la facteur-representation d'un groupe de Lie rgsoluble de Type (E), Math. J. Okayama Univ. 7 (1957), 151-161.

Chat View PDF

O-minimal structures

2007-11-01

A. J. Wilkie

The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries [vdD1] as a framework for investigating the model theory of … The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries [vdD1] as a framework for investigating the model theory of the real exponential function exp : R ! R : x ! ex, and thereby settle an old problem of Tarski. More on this later, but for the moment it is best motivated as being a candidate for Grothendieck’s idea of “tame topology” as expounded in his Esquisse d’un Programme [Gr]. It seems to me that such a candidate should satisfy (at least) the following criteria. (A) It should be a framework that is flexible enough to carry out many geometrical and topological constructions on real functions and on subsets of real euclidean spaces. (B) But at the same time it should have built in restrictions so that we are a priori guaranteed that pathological phenomena can never arise. In particular, there should be a meaningful notion of dimension for all sets under consideration and any that can be constructed from these by use of the operations allowed under (A). (C) One must be able to prove finiteness theorems that are uniform over fibred collections. None of the standard restrictions on functions that arise in elementary real analysis satisfy both (A) and (B). For example, there exists a continuous function G : (0, 1) ! (0, 1)2 which is surjective, thereby destroying any hope of a dimension theory for a framework that admits all continuous functions. Restricting to the smooth (i.e. C1) 985–02 environment fares no better. For every closed subset of any euclidean space, in particular the subset graph(G) of R3, is the set of zeros of some smooth function. So by the use of a few simple constructions that we would certainly wish to allow under (A), we soon arrive at dimension-destroying phenomena. The same is even true (though this is harder to prove) if we start from just those smooth functions that are everywhere real analytic (i.e. equal the sum of their Taylor series on a neighbourhood of every point), although, as we shall see, this class of functions is locally well-behaved and as such can serve as a model for the three criteria above. Rather than enumerate analytic conditions on sets and functions sufficient to guarantee the criteria (A), (B) and (C) however, we shall give one succinct axiom, the o-minimality axiom, which implies them. Of course, this is a rather open-ended (and currently flourishing) project because of the large number of questions that one can ask under (C). One must also provide concrete examples of collections of sets and functions that satisfy the axiom and this too is an active area of research. In this talk I shall survey both aspects of the theory. Our formulation of the o-minimality axiom makes use of definability theory from mathematical logic. We begin with a collection F of real valued functions of real variables (not necessarily all of the same number of arguments). We consider the ordered field structure on R augmented by the functions in F. This gives us a first-order structure (or model ) RF := hR;+, ·,−,<,Fi, and we denote the corresponding firstorder logical language by L(F). We then call the structure RF o-minimal if whenever (x) is an L(F)-formula (with parameters) then the subset of R defined by (x) is a finite union of open intervals and points (i.e. it is the union of finitely many connected sets). I shall elucidate what is meant by an L(F)-formula and by the subset of R (and, more generally, of Rn) defined by such a formula in the next two sections. However, I should emphasize at this stage that such a formula not only defines a subset , denoted (RF), of Rn, but also a subset (R) of Rn where R is any ordered ring augmented by a collection of functions, F say, such that F and F are in correspondence via a bijection that preserves the number of places (arity) of the functions. One can, and should, define the notion o-minimality for such structures hR;Fi and it was at (rather more than) this level of generality that the true foundations of the subject were laid by Pillay and Steinhorn in [P-S], shortly after van den Dries’ work on the real field. Indeed, it turned out that the solution to Tarski’s problem on the real exponential function (the case F = {exp} in the above notation) relied heavily on the Pillay-Steinhorn theory of o-minimality for structures based on ordered fields other than the reals. This having been said, I shall concentrate in this lecture on the real case, alluding only occasionally to the more general situation, and leave the reader to adapt the definitions and theorems to the setting of o-minimal expansions of arbitrary ordered fields.

Solvable groups of finite Morley rank

1989-02-01

Ali Nesin

A Theorem on compact semi-simple groups

1949-12-01

Morikuni Gotô

Chat View PDF

Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

Chat View PDF

Linear Groups Definable in o-Minimal Structures

2002-01-01

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

On commutators in real semisimple Lie groups

1986-03-01

Dragomir Z̆. Ðoković

Cosets, genericity, and the Weyl group

2009-06-03

Éric Jaligot

Chat View PDF

Generix never gives up

2006-06-01

Éric Jaligot

Abstract We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal … Abstract We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal counterexample to the Genericity Conjecture.

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22

Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Groups of Finite Morley Rank with Solvable Local Subgroups

2012-02-29

Adrien Deloro , Éric Jaligot

Abstract We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these … Abstract We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove a main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases and study torsion. Key Words: Bender methodCherlin–Zilber conjectureFeit–Thompson theoremMorley rank2000 Mathematics Subject Classification: Primary 20F11Secondary 20E25, 20E34 Notes Communicated by D. Macpherson.

Chat View PDF

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.

Class Functions, Conjugacy Classes and Commutators in Semisimple Lie Groups

2001-01-01

Armand Borel

LIE GROUPS AND ALGEBRAIC GROUPS

1991-11-01

Andrew Pressley

By A. L. Onishchik and E. B. Vinberg (translated from the Russian by D. A. Leites): 328 pp., DM.128.–, ISBN 3 540 50614 4 (Springer, 1990). By A. L. Onishchik and E. B. Vinberg (translated from the Russian by D. A. Leites): 328 pp., DM.128.–, ISBN 3 540 50614 4 (Springer, 1990).

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).

On central extensions and definably compact groups in o-minimal structures

2010-11-17

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

Chat View PDF

Locally finite groups

1973-01-01

Otto H. Kegel , B. A. F. Wehrfritz