Real Lie groups and o-minimality

Authors

Annalisa Conversano, Alf Onshuus, Sacha Post

Type: Article

Publication Date: 2021-09-22

Citations: 0

DOI: https://doi.org/10.1090/proc/15847

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Abstract

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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Proceedings of the American Mathematical Society
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Summary

Real Lie groups and o-minimality

2021-01-01

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. We characterize, up to Lie isomorphism, the real Lie groups that are definable in an o-minimal expansion of the real field.

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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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Solvable Lie groups definable in o-minimal theories

2015-01-01

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.

Solvable Lie groups definable in o-minimal theories

2015-06-26

Annalisa Conversano , Alf Onshuus , Sergei Starchenko

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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A definability criterion for linear Lie groups

2019-10-24

Alf Onshuus , Sacha Post

It is known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to … It is known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is then natural to ask when does a Lie group is Lie isomorphic to a group definable in such expansion. Conversano, Starchenko and the first author answered this question in \cite{COSsolvable} in the case where the group is solvable. We give here a criterion in the case where the group is linear. More precisely if $G$ is a linear Lie group it is isomorphic to a group definable in an $o$-minimal expansion of the reals if and only if its solvable radical is isomorphic to such group.

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.

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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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

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).

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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.

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.

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A definability criterion for connected Lie groups

2019-01-01

Alf Onshuus , Sacha Post

It has been known since \cite{Pgroupchunk} that any group definable in an $o$-minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural … It has been known since \cite{Pgroupchunk} that any group definable in an $o$-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 \cite{COSsolvable} 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 $G$ is Lie isomorphic to a group definable in an $o$-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 $G$ has a "good Levi descomposition", we prove that in order for $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in \cite{COSsolvable}.

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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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The regular group 𝐶*-algebra for real-rank one groups

1976-01-01

Robert P. Boyer , Robert P. Martin

Let <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> be a connected semisimple real-rank one Lie group with finite center and let <inline-formula content-type="math/mathml"> <mml:math … Let <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> be a connected semisimple real-rank one Lie group with finite center and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript rho Superscript asterisk Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>ρ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_\rho ^{\ast }(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the regular group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{C^{\ast }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra 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>. In this paper a complete description of the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript rho Superscript asterisk Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>ρ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_\rho ^{\ast }(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained.

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Infinite Lie rings

1956-01-01

Haruo Sunouchi

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

2019-01-01

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

2019-04-22

Annalisa Conversano

Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

2024-07-01

Alexandru Chirvăsitu

An extended derivation (endomorphism) of a (restricted) Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an assignment of a derivation (respectively) of … An extended derivation (endomorphism) of a (restricted) Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an assignment of a derivation (respectively) of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L prime"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">L’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any (restricted) Lie morphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper L right-arrow upper L prime"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f:L\to L’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, functorial in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L prime"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">L’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and (b) if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms.

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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.

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 .

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Definable homomorphisms of abelian groups in o-minimal structures

1999-11-01

Ya’acov Peterzil , Sergei Starchenko

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01

Margarita Otero , Ya’acov Peterzil , Anand Pillay

Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Automorphisms of group extensions

1971-01-01

Charles Wells

If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j> of E which takes G onto itself induces automorphisms t on … If 1 ->■ G -í> E^-Tt -> 1 is a group extension, with i an inclusion, any automorphism <j> of E which takes G onto itself induces automorphisms t on G and a on n.However, for a pair (a, t) of automorphism of n and G, there may not be an automorphism of E inducing the pair.Let à: n -*■ Out G be the homomorphism induced by the given extension.A pair (a, t) e Aut n x Aut G is called compatible if a fixes ker á, and the automorphism induced by a on Hü is the same as that induced by the inner automorphism of Out G determined by t.Let C< Aut IT x Aut G be the group of compatible pairs.Let Aut (E; G) denote the group of automorphisms of E fixing G.The main result of this paper is the construction of an exact sequence 1 -» Z&T1, ZG) -* Aut (E; G)-+C^ H*(l~l, ZG).The last map is not surjective in general.It is not even a group homomorphism, but the sequence is nevertheless "exact" at C in the obvious sense.

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Commutators in groups definable in o-minimal structures

2012-02-27

Elías Baro , Éric Jaligot , Margarita Otero

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 … 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.

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