Connected components of definable groups, and o-minimality II

Conversano, Annalisa; Pillay, Anand

doi:10.1016/j.apal.2015.04.005

Cited by 28 publications

(68 citation statements)

References 25 publications

(65 reference statements)

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“…In [3], the authors have shown that such extensions can, under some additional assumptions, give new examples of definable groups with G 00 = G 000 , building upon and extending the intuitions from the first known example with this property, found in [5], namely the universal cover of SL 2 (R). They also pose some questions and conjectures, one of which will be proved at the end of this section.…”

Section: ( † †)mentioning

confidence: 93%

Smoothness of Bounded Invariant Equivalence Relations

Krupiński¹,

Rzepecki²

2016

J. symb. log.

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We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion. §1. Introduction.1.1. Preface. This paper will concern the Borel cardinalities of bounded, invariant equivalence relations, as well as some weak analogues in an uncountable case. More precisely, we are concerned with the connection between typedefinability and smoothness of these relations-type-definable equivalence relations are always smooth (cf. Fact 2.14), while the converse is not true in general. We also apply this to the study of connected components in definable group extensions.The general motivation for the use of Borel cardinality in the context of bounded invariant equivalence relations is a better understanding of "spaces" of strong types (i.e., "spaces" of classes of such relations). For a bounded type-definable equivalence relation, its set of classes, equipped with the so-called logic topology, forms a compact Hausdorff topological space. However, for relations which are only invariant, but not type-definable, the logic topology is not necessarily Hausdorff, so it is not so useful. The question arises how to measure the complexity of the spaces of classes of such relations. One of the ideas is to investigate their Borel cardinalities, which was formalised in [4], wherein the authors asked whether the Lascar strong type must be nonsmooth if it is not equal to the Kim-Pillay strong type. This question was answered in the positive in [4], and in this paper, we generalise its methods to a more general class of invariant equivalence relations, and we find an important application in the context of definable group extensions.

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Section: ( † †)mentioning

confidence: 93%

Smoothness of Bounded Invariant Equivalence Relations

Krupiński¹,

Rzepecki²

2016

J. symb. log.

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“…And Proposition 2.1(i) of [4] says that H is solvable and there are definable 1 = H 0 < ... < H n = H where each H i+1 /H i is 1-dimensional, torsion-free (definably connected). Also by Proposition 4.7 of [4] H has a global left invariant definable (f -generic) type, so H fits into one of the extreme cases in the previous section.…”

Section: O-minimalitymentioning

confidence: 99%

“…We have the following decomposition theorem [4], Proposition 4.6: Fact 1.18. Suppose G is definably connected.…”

Section: O-minimalitymentioning

confidence: 99%

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On minimal flows, definably amenable groups, and o-minimality

Pillay

Yao

2016

Advances in Mathematics

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“…The universal covering group G of a definably connected group G that is interpretable in an o-minimal expansion M of the field R is a NIP group: G is interpretable in the two sorted structure (( 1 (G), +), M ) (E. Hrushovski et al [13]) hence NIP (A. Conversano and A. Pillay [6]). An ultraproduct of groups that are uniformly interpretable in a NIP structure is NIP (D. Macpherson and K. Tent [19]).…”

Section: ) There Is a Nice Subgroup H ⊂ G With S ⊂ H Such That H Is mentioning

confidence: 99%

Variations Sur Un Thème De Aldama Et Shelah

Milliet

2016

J. symb. log.

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Abstract. We consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length is contained in an S-invariant externally definable soluble subgroup of G of derived length . The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length . Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

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Connected components of definable groups, and o-minimality II

Cited by 28 publications

References 25 publications

Smoothness of Bounded Invariant Equivalence Relations

Smoothness of Bounded Invariant Equivalence Relations

On minimal flows, definably amenable groups, and o-minimality

Variations Sur Un Thème De Aldama Et Shelah

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