On torsion-free groups in o-minimal structures

Authors

Ya’acov Peterzil, Sergei Starchenko

Type: Article

Publication Date: 2005-10-01

Citations: 27

DOI: https://doi.org/10.1215/ijm/1258138139

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Abstract

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 .

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Illinois Journal of Mathematics
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On the homotopy type of definable groups in an o-minimal structure

2011-02-24

Alessandro Berarducci , Marcello Mamino

We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 … We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 → G00 → G → G/G00 → 1, where G00 is the ‘infinitesimal subgroup’ of G and G/G00 is a compact real Lie group. We show that given a connected open subset U of G/G00, there is a canonical isomorphism between the fundamental group of U and the o-minimal fundamental group of its preimage under the projection p: G→ G/G00. We apply this result to show that there is a natural exact sequence 1 → G 00 → G ~ → G / G 00 ~ → 1 , where G ~ is the (o-minimal) universal cover of G, and G / G 00 ~ is the universal cover of the real Lie group G/G00. We also prove that, up to isomorphism, each finite covering H → G/G00, with H a connected Lie group, is of the form H/H00→ G/G00 for some definable group extension H→G. Finally we prove that the (Lie-)isomorphism type of G/G00 determines the definable homotopy type of G. In the semisimple case a stronger result holds: G/G00 determines G up to definable isomorphism. Our results depend on the study of the o-minimal fundamental groupoid of G and the homotopy properties of the projection G→ G/G00.

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Linear Groups Definable in o-Minimal Structures

2002-01-01

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

Topology of definable abelian groups in o-minimal structures

2011-11-17

Elías Baro , Alessandro Berarducci

In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to … In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.

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On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

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.

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

2011-01-01

Annalisa Conversano , Anand Pillay

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A REDUCTION TO THE COMPACT CASE FOR GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

2014-03-01

Annalisa Conversano

Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal … Abstract Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M . We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K , which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$ . It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$ ), and homotopy equivalent to K . This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.

Strongly minimal groups in o-minimal structures

2021-05-28

Pantelis E. Eleftheriou , Assaf Hasson , Ya’acov Peterzil

We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle … We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle a 2-dimensional group definable in \mathcal{M} , and \mathcal{D}=\langle G;+,\ldots\rangle a strongly minimal structure, all of whose atomic relations are definable in \mathcal{M} . If \mathcal{D} is not locally modular, then an algebraically closed field K is interpretable in \mathcal{D} , and the group G , with all its induced \mathcal{D} -structure, is definably isomorphic in \mathcal{D} to an algebraic K -group with all its induced K -structure.

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Definability of continuous isomorphisms of groups definable in o-minimal expansions of the real field

2023-01-01

Alf Onshuus

In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer … In this paper, we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field (which we will refer to as "definable groups"). It is known (\cite{Pi88}) that any group definable in an o-minimal expansion of the real field is a Lie group, and in \cite{COP} a complete characterization of when a Lie group has a "definable group" which is \emph{Lie isomorphic} to it was given. We continue the analysis by explaining when a Lie homomorphism between definable groups is a definable isomorphism. Among other things, we prove that in any o-minimal expansion $\mathcal R$ of the real field we can add a function symbol for any Lie isomorphism between definable groups to the language of $\mathcal R$ preserving o-minimality, and that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.

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Zero-groups and maximal tori

2005-01-01

Alessandro Berarducci

We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact … We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.

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DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

Mário J. Edmundo , Margarita Otero

Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the … Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤ n ; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ) n , and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.

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Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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Solvable groups definable in o-minimal structures

2000-01-01

Mário J. Edmundo

Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove … Let N be an o-minimal structure. In this paper we develop group extension and group cohomology theory over N and use it to describe the N-definable solvable groups. We prove an o-minimal analogue of the Lie-Kolchin-Mal'cev theorem and we describe the N-definable G-modules and the N-definable rings.

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GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

2008-01-01

Pantelis E. Eleftheriou

In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure … In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups deflnable in o-minimal structures, and c) present a structure theorem for the special case of semi-linear groups, exemplifying their relation with real Lie groups.

On the o-minimal Hilbert's fifth problem

2015-01-01

Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal … Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k>0$, the $k$-torsion subgroups of $G;$ the o-minimal cohomology algebra over ${\mathbb Q}$ of $G.$ As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

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Locally definable groups in o-minimal structures

2005-05-18

Mário J. Edmundo

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Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01

Ya’acov Peterzil , Sergei Starchenko

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

2013-01-01

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 a rather strong definition of <i>ind-definable semisimple subgroup</i>,

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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.

Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2021-08-19

Annalisa Conversano

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Connected components of definable groups and o-minimality I

2011-01-29

Annalisa Conversano , Anand Pillay

We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is … We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.

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.

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Interpretable groups are definable

2014-03-18

Pantelis E. Eleftheriou , Ya’acov Peterzil , Janak Ramakrishnan

We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product … We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.

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Interpretable groups are definable

2011-10-30

Janak Ramakrishnan , Ya’acov Peterzil , Pantelis E. Eleftheriou

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Locally definable subgroups of semialgebraic groups

2018-12-27

Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic … We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.

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Locally definable subgroups of semialgebraic groups

2019-10-07

Elías Baro , Pantelis E. Eleftheriou , Ya’acov Peterzil

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: … We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2021-08-19

Annalisa Conversano

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Connected components of definable groups, and o-minimality II

2015-05-04

Annalisa Conversano , Anand Pillay

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Classifying torsion free groups in o-minimal expansions of real closed fields

2016-07-15

Eliana Barriga , Alf Onshuus

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

2019-02-26

Elías Baro

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Solvable Lie groups definable in o-minimal theories

2015-06-26

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.

Fusing o-minimal structures

2005-03-01

A. J. Wilkie

Abstract In this note I construct a proper o-minimal expansion of the ordered additive group of rationals. Abstract In this note I construct a proper o-minimal expansion of the ordered additive group of rationals.

Definable topological dynamics and real Lie groups

2015-02-01

Grzegorz Jagiella

We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, … We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition , we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to . This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to . We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort and a sort for an abelian group.

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.

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Connected components of definable groups and <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>-minimality I

2012-06-29

Annalisa Conversano , Anand Pillay

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Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups

2021-01-01

Eliana Barriga

We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application … We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Thm. 9.1] (see Corollary 2.2). We also prove that Theorem 10.2 in [3] also holds for any definably connected definably compact semialgebraic group $G$ not necessarily abelian over a sufficiently saturated real closed field $R$; namely, that the o-minimal universal covering group $\widetilde{G}$ of $G$ is an open locally definable subgroup of $\widetilde{H\left(R\right)^{0}}$ for some $R$-algebraic group $H$ (Thm. 3.3). Finally, for an abelian definably connected semialgebraic group $G$ over $R$, we describe $\widetilde{G}$ as a locally definable extension of subgroups of the o-minimal universal covering groups of commutative $R$-algebraic groups (Theorem 3.4)

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Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

2019-04-22

Annalisa Conversano

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite … We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.

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Decomposable ordered groups

2014-01-01

Eliana Barriga , Alf Onshuus , Charles Steinhorn

Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a … Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.

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Topology of definable abelian groups in o-minimal structures

2011-11-17

Elías Baro , Alessandro Berarducci

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Groups definable in linear O-minimal structures

2007-06-29

Pantelis E. Eleftheriou

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

2018-02-20

Elías Baro

We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the … We show that the derived subgroup of a linear definable group in an o-minimal structure is also definable, extending the semialgebraic case proved by A. Pillay. We also show the definability of the derived subgroup in case that the group is simply-connected.

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.

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MAXIMAL COMPACT SUBGROUPS IN THE O-MINIMAL SETTING

2013-04-11

Annalisa Conversano

A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role … A characterization of groups definable in o-minimal structures having maximal definable definably compact subgroups is given. This follows from a definable decomposition in analogy with Lie groups, where the role of maximal tori is played by maximal 0-subgroups. Along the way we give structural theorems for solvable groups, linear groups, and extensions of definably compact by torsion-free definable groups.

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.

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Extensions of definable local homomorphisms in o‐minimal structures and semialgebraic groups

2024-07-17

Eliana Barriga

Abstract We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of … Abstract We state conditions for which a definable local homomorphism between two locally definable groups , can be uniquely extended when is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group not necessarily abelian over a sufficiently saturated real closed field ; namely, that the o‐minimal universal covering group of is an open locally definable subgroup of for some ‐algebraic group (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group over , we describe as a locally definable extension of subgroups of the o‐minimal universal covering groups of commutative ‐algebraic groups (Theorem 3.4).

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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:

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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.

O-minimal cohomology and definably compact definable groups

2000-01-01

Mário J. Edmundo

Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the … Let N be an o-minimal expansion of a real closed field. We develop cohomology theory for the category of N-definable manifolds and N-definable maps, and use this to solve the Peterzil-Steinhorn problem on the existence of torsion points on N-definably compact N-definable abelian groups. We compute the cohomology rings of N-definably compact N-definable groups, and we prove an o-minimal analog of the Poincare duality theorem, the Alexander dualti theorem, the Lefschetz duality theorem and the Lefschetz fixed point theorem.

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A growth dichotomy for o-minimal expansions of ordered groups

1998-01-01

Chris Miller , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an o-minimal expansion of a divisible ordered abelian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,+,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a distinguished positive element <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the following dichotomy holds: Either there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable binary operation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dot"> <mml:semantics> <mml:mo>⋅</mml:mo> <mml:annotation encoding="application/x-tex">\cdot</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma plus comma dot comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,+,\cdot ,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an ordered real closed field; or, for every definable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper R right-arrow upper R"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:R\to R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exists a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda element-of StartSet 0 EndSet union upper A u t left-parenthesis upper R comma plus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>∪</mml:mo> <mml:mi>Aut</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda \in \{0\}\cup \operatorname {Aut}(R,+)</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="limit Underscript x right-arrow plus normal infinity Endscripts left-bracket f left-parenthesis x right-parenthesis minus lamda left-parenthesis x right-parenthesis right-bracket element-of upper R"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:munder> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>λ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M colon equals left-parenthesis upper M comma greater-than comma ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:mo>:=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}:=(M,>,\dots )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable groups with underlying set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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The Topology of Fibre Bundles.

1951-03-21

Norman Steenrod

Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds and covering spaces. … Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds and covering spaces. It provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles.

A growth dichotomy for o-minimal expansions of ordered fields

1996-08-22

Chris Miller

Algebraic Groups and Class Fields

1988-01-01

Jean-Pierre Serre

The real field with convergent generalized power series

1998-01-01

Lou van den Dries , Patrick Speissegger

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on $[0,1]$ by a series $\sum c_{n} x^{\alpha _{n}}$ with … We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on $[0,1]$ by a series $\sum c_{n} x^{\alpha _{n}}$ with $0 \leq \alpha _{n} \rightarrow \infty$ and $\sum |c_{n}| r^{\alpha _{n}} < \infty$ for some $r>1$ is definable. This expansion is polynomially bounded.

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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.

Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01

Ya’acov Peterzil , Charles Steinhorn

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

One-dimensional groups over an o-minimal structure

1991-09-01

Vladimir Razenj

Expansions of the real field with power functions

1994-06-01

Chris Miller

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

One-dimensional groups definable in o-minimal structures

1994-10-01

Adam Strzeboński

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Reducts of some structures over the reals

1993-09-01

Ya’acov Peterzil

Abstract We consider reducts of the structure ℛ = 〈ℝ, +, ·, <〉 and other real closed fields. We compete the proof that there exists a unique reduct between 〈ℝ, … Abstract We consider reducts of the structure ℛ = 〈ℝ, +, ·, <〉 and other real closed fields. We compete the proof that there exists a unique reduct between 〈ℝ, +, <,λ a 〉 a ∈ ℝ and ℛ, and we demonstrate how to recover the definition of multiplication in more general contexts than the semialgebraic one. We then conclude a similar result for reducts between 〈ℝ, ·, <〉 and ℛ and for general real closed fields.

An application of model theory to real and p-adic algebraic groups

1989-10-01

Anand Pillay

Euler characteristic in semialgebraic and other o-minimal groups

1994-10-01

Adam Strzeboński

Solvable groups definable in o-minimal structures

2003-10-09

Mário J. Edmundo

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Linear Groups Definable in o-Minimal Structures

2002-01-01

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

The Field of Reals with Multisummable Series and the Exponential Function

2000-11-01

Lou van den Dries , Patrick Speissegger

We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model … We show that the field of real numbers with multisummable real power series is model complete, o-minimal and polynomially bounded. Further expansion by the exponential function yields again a model complete and o-minimal structure which is exponentially bounded, and in which the Gamma function on the positive real line is definable. 2000 Mathematics Subject Classification: primary 03C10, 32B05, 32B20; secondary, 26E05.

Transfer methods for o-minimal topology

2003-09-01

Alessandro Berarducci , Margarita Otero

Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties … Abstract Let M be an o-minimal expansion of an ordered field. Let φ be a formula in the language of ordered domains. In this note we establish some topological properties which are transferred from φ M to φ R and vice versa. Then, we apply these transfer results to give a new proof of a result of M . Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

DEFINABLY COMPACT ABELIAN GROUPS

2004-12-01

Mário J. Edmundo , Margarita Otero

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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.

Some Basic Theorems on Algebraic Groups

1956-04-01

Maxwell Rosenlicht

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received … The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

Extensions of Vector Groups by Abelian Varieties

1958-07-01

Maxwell Rosenlicht

Extending Tamm's theorem

1994-01-01

Lou van den Dries , Chris Miller

Let f:A→ℝ,A⊆ℝ m+n , be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions x↦x r :(0,∞)→ℝ,r∈ℝ. Then … Let f:A→ℝ,A⊆ℝ m+n , be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions x↦x r :(0,∞)→ℝ,r∈ℝ. Then there exists N∈ℕ such that for all (a,b)∈A, if y↦f(a,y) is C N in a neighborhood of b, then y↦f(a,y) is real analytic in a neighborhood of b.

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Lie Groups and Lie Algebras

1998-01-01

Boris Komrakov , I. S. Krasil’shchik , G. L. Litvinov +1 more

Definable Sets in Ordered Structures. I

1986-06-01

Anand Pillay , Charles Steinhorn

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly … This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly ©-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories.Theorems 2.1 and 2.3, respectively, provide characterizations of C-minimal ordered groups and rings.Several other simple results are collected in §3.The primary tool in the analysis of ¿¡-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2.This result states that any (parametrically) definable unary function in an (5-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals.The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0-categorical ¿¡¡-minimal structures (Theorem 6.1).

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