On model-theoretic connected components in some group extensions

Gismatullin, Jakub; Krupiński, Krzysztof

doi:10.1142/s0219061315500099

Cited by 10 publications

(47 citation statements)

References 23 publications

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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%

“…The above conjectures are important mostly for two reasons: (1) They imply that Corollary 2.8 in [3] holds in general (i.e., also when the language is uncountable), that is: in Theorem 5.5, if G 00 B = G 000 B and A * 1 ⊆ G 000 B ∩ A, then the assumption (i) (about the nonsplitting of the modified 2-cocycle) not only implies (under the assumption (ii)), but is also necessary for G 00 B = G 000 B . This is explained in detail in [3]. (2) They imply that, in a rather general context, the quotient G 00 / G 000 is (algebraically) isomorphic to the quotient of a compact group by a finitely generated dense subgroup (this will be revisited at the end of this section).…”

Section: Application Of the Main Theorem In Group Extensionsmentioning

confidence: 99%

“…If we assume, in addition, that E refines ≡ KP (on X ), then conditions (2), (3),(4) are equivalent (and equivalent to simply E = ≡ KP X ) and are implied by, but do not imply (1).…”

Section: • (2) Implies (3) and (4) But Not (1) • (3) Is Equivalent mentioning

confidence: 99%

“…If, in the first part of Theorem 4.9, we drop the assumption that E is orbital on types (so we allow E to not refine type), then (2) does not imply (4)-as witnessed by Example 2.25 (though (1) certainly still implies the other conditions and (2) implies (3) and that the classes are pseudoclosed).…”

Section: Possible Extensions Of the Characterisation Theoremmentioning

confidence: 99%

“…(3) Theorem 4.9, in which we attempt to analyse in detail the connection between smoothness, type-definability and some other properties of bounded and invariant equivalence relations, under some additional assumptions; it uses a corollary of Theorem 3.4 to show that some of these properties are stronger than others, and several (original) examples to show that they are not equivalent. (4) Theorem 5.6, which applies Corollary 3.19 along with some ideas from [3] and [4]. in the context of definable group extensions, in order to give a criterion for type-definability of subgroups of such extensions, resulting in a proof of important technical conjectures (see Conjectures 5.9 and 5.10 in the last section) from [3] in Corollary 5.15; the motivation for these conjectures is recalled in the remark following them.…”

mentioning

confidence: 99%

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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%

Section: Application Of the Main Theorem In Group Extensionsmentioning

confidence: 99%

“…If we assume, in addition, that E refines ≡ KP (on X ), then conditions (2), (3),(4) are equivalent (and equivalent to simply E = ≡ KP X ) and are implied by, but do not imply (1).…”

Section: • (2) Implies (3) and (4) But Not (1) • (3) Is Equivalent mentioning

confidence: 99%

Section: Possible Extensions Of the Characterisation Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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Smoothness of Bounded Invariant Equivalence Relations

Krupiński¹,

Rzepecki²

2016

J. symb. log.

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Definable topological dynamics and real Lie groups

Jagiella

2015

Mathematical Logic Qtrly

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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 G=KH, we give a description of minimal subflows and the Ellis group of its universal definable flow SG(double-struckR) in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to NG(H)∩K(double-struckR). This provides a range of counterexamples to a question by Newelski whether the Ellis group is isomorphic to G/G00. We further extend the results to universal topological covers of definable groups, interpreted in a two‐sorted structure containing the o‐minimal sort double-struckR and a sort for an abelian group.

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Amenability, connected components, and definable actions

Hrushovski

Krupiński

Pillay

2021

Sel. Math. New Ser.

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We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

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On model-theoretic connected components in some group extensions

Cited by 10 publications

References 23 publications

Smoothness of Bounded Invariant Equivalence Relations

Smoothness of Bounded Invariant Equivalence Relations

Definable topological dynamics and real Lie groups

Amenability, connected components, and definable actions

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