Invariance results for definable extensions of groups

Edmundo, Mário J.; Jones, Gareth A.; Peatfield, Nicholas J.

doi:10.1007/s00153-010-0196-5

Cited by 6 publications

(13 citation statements)

References 24 publications

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“…(⇒): Via [17], G is definably equipped with the structure of a real Lie group and π : G → H is a (definable) homomorphism of Lie groups, and as the kernel is finite, must be a topological covering. (⇐) appears in [13] (see Theorems 2.8 and 8.4) as well as in [9] (Theorem 1.4).…”

Section: Proposition 22 Assume M Is An O-minimal Expansion Of the Rmentioning

confidence: 95%

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GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

2010

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We study the model theory of "covers" of groups H definable in an o-minimal structure M . We pose the question of whether any finite central extension G of H is interpretable in M , proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω 1 ,ω -categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).

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Section: Proposition 22 Assume M Is An O-minimal Expansion Of the Rmentioning

confidence: 95%

“…For example 2 =⇒ 1 can be seen to follow from Theorem 8.2 of [13]. And 3 =⇒ 1 can be seen as a restatement of Corollary 1.2 of [9]. Neverthless for the benefit of the reader we will give more or less direct proofs, following a sequence of lemmas.…”

Section: Stable Embeddednessmentioning

confidence: 99%

GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

2010

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“…The following is the main theorem in this section. It appears to be a corollary of Theorem 1.4 in [11], but our proof is constructive so we decided to include it. Theorem Let

H\longrightarrow G

be a finite covering map of a connected definable Lie group

G

.…”

Section: Finite Covers Of Definable Lie Groupsmentioning

confidence: 99%

A definability criterion for connected Lie groups

Onshuus

Post

2022

Bulletin of London Math Soc

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It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239-255)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 (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 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 its Levi subgroups with finite center, 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 (Conversano, Onshuus,

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“…In fact, besides (P1) and (P2) (and their consequences) everything else that is required is, on the one hand, results from [18], which hold in arbitrary o-minimal structures (and for locally definable spaces as well), and on the other hand, [10, Chapter 6, (3.6)], which is used to notice that the domains of the "good" definable paths are definably normal. In our case here the good definable paths are the definable J-paths and their domains are Hausdorff, definably compact definable spaces (Lemma 2.5), definable in the o-minimal structure J with definable choice (Remark 3.6) and so they are definably normal by [19,Theorem 2.11].…”

mentioning

confidence: 99%

“…In the o-minimal context (here and in [16]), the role that (P1) (b) and (P2) play is similar to the role the analogue properties play in topology. However, (P2) is often used in combination with the results from [18] mentioned above to get local definability. Also (P1) (a) is required essentially only once and to get local definability (see [16,Proposition 2.18]), the other places where it is used, it is used to replace definably connected by definably path connected.…”

mentioning

confidence: 99%

On Pillay's conjecture in the general case

Edmundo

Mamino

Prelli

et al. 2017

Advances in Mathematics

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Let M be an arbitrary o-minimal structure. Let G be a definably compact, definably connected, abelian definable group of dimension n. Here we compute: (i) the new intrinsic o-minimal fundamental group of G; (ii) for each k > 0, the k-torsion subgroups of G; (iii) the o-minimal cohomology algebra over 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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Invariance results for definable extensions of groups

Cited by 6 publications

References 24 publications

GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

A definability criterion for connected Lie groups

On Pillay's conjecture in the general case

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