Cartan subgroups of groups definable in o-minimal structures

Baro, Elías; Jaligot, Eric; Otero, Margarita

doi:10.1017/s1474748013000352

Cited by 3 publications

(31 citation statements)

References 32 publications

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“…We survey some preliminary results proved in for o‐minimal structures (not necessarily expansions of real closed fields). Fact Let

G

be a group definable in an o‐minimal structure

scriptM

.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

“…Fact Let

G

be a group definable in an o‐minimal structure

scriptM

. Then, (1) [ , Theorem 1] Cartan subgroups of

G

exist, are definable in

scriptM

and they fall into finitely many conjugacy classes, and (2) [ , Corollary 75] if

G

is definably connected and

H

is a Cartan subgroup of

G

, then

H = C_{G} false(H^{o} false) H^{o}

, in particular if

H_{1}

and

H_{2}

are Cartan subgroups of

G

, then

H_{1}^{o} = H_{2}^{o}

implies

H_{1} = H_{2}

.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

“…Remark Let

G

be a group definable in an o‐minimal structure

scriptM

. For each Carter subgroup

Q

G

H : = C_{G} false(Q false) Q

is the unique Cartan subgroup of

G

containing

Q

by [, Lemma 5]. The definably connected component of a Cartan subgroup is a Carter subgroup.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

“…In § 3, we answer positively the main questions left open in . The study of Cartan subgroups of groups definable in o‐minimal structures was initiated in , where it was proven that Cartan subgroups of a definable group exist, are definable, and fall into finite many conjugacy classes [, Theorem 1].…”

Section: Introductionmentioning

confidence: 95%

“…The study of Cartan subgroups of groups definable in o‐minimal structures was initiated in , where it was proven that Cartan subgroups of a definable group exist, are definable, and fall into finite many conjugacy classes [, Theorem 1]. In Corollary , we prove, among other results, that the union

X

of the Cartan subgroups of a given definable group

G

is a dense subset of

G

(see [, Question 84]). The latter implies that the set

X

is syndetic in

G

(also called ‘generic') that is, there is a finite

E \subseteq G

such that

X E = G

, and we conjecture (Conjecture ) that it suffices to consider the conjugates of a specially chosen Cartan subgroup to get a syndetic subset of

G

.…”

Section: Introductionmentioning

confidence: 99%

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Cartan subgroups and regular points of o‐minimal groups

Baro

Berarducci

Otero

2019

J. London Math. Soc.

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Let G be a group definable in an o‐minimal structure scriptM. We prove that the union of the Cartan subgroups of G is a dense subset of G. When scriptM 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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“…We survey some preliminary results proved in for o‐minimal structures (not necessarily expansions of real closed fields). Fact Let

G

be a group definable in an o‐minimal structure

scriptM

.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

“…Fact Let

G

be a group definable in an o‐minimal structure

scriptM

. Then, (1) [ , Theorem 1] Cartan subgroups of

G

exist, are definable in

scriptM

and they fall into finitely many conjugacy classes, and (2) [ , Corollary 75] if

G

is definably connected and

H

is a Cartan subgroup of

G

, then

H = C_{G} false(H^{o} false) H^{o}

, in particular if

H_{1}

and

H_{2}

are Cartan subgroups of

G

, then

H_{1}^{o} = H_{2}^{o}

implies

H_{1} = H_{2}

.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

“…Remark Let

G

be a group definable in an o‐minimal structure

scriptM

. For each Carter subgroup

Q

G

H : = C_{G} false(Q false) Q

is the unique Cartan subgroup of

G

containing

Q

by [, Lemma 5]. The definably connected component of a Cartan subgroup is a Carter subgroup.…”

Section: Cartan Subgroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

X

of the Cartan subgroups of a given definable group

G

is a dense subset of

G

(see [, Question 84]). The latter implies that the set

X

is syndetic in

G

(also called ‘generic') that is, there is a finite

E \subseteq G

such that

X E = G

, and we conjecture (Conjecture ) that it suffices to consider the conjugates of a specially chosen Cartan subgroup to get a syndetic subset of

G

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cartan subgroups and regular points of o‐minimal groups

Baro

Berarducci

Otero

2019

J. London Math. Soc.

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Quasi groupes de Frobenius dimensionnels

Zamour

2023

Model Th.

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Nous nous intéressons à une classe de groupes, les quasi groupes de Frobenius (avec involutions), dont la structure interne généralise celle des groupes classiques GA 1 ‫,)ރ(‬ PGL 2 ‫)ރ(‬ et SO 3 ‫)ޒ(‬ : 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 de 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.We are interested in a class of groups, quasi Frobenius groups (with involutions), whose internal structure generalizes that of the classical groups GA 1 ‫,)ރ(‬ PGL 2 ‫)ރ(‬ and SO 3 ‫:)ޒ(‬ a subgroup and its conjugates, of finite index in their normalizer and trivial mutual intersection, cover "generically" the ambient group. From the perspective of model theory, we work with the hypothesis of the existence of a good notion of dimension on definable sets (we must distinguish between the o-minimal case and the ranked case). We pay special attention to the ranked case. By studying the geometry of incidence induced by involutions, we sketch a classification of quasi Frobenius groups and thus determine under which conditions classical groups can be identified in a dimensional framework.

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