Results 1  10
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33
On NIP and invariant measures
, 2007
"... We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the com ..."
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Cited by 43 (14 self)
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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of “generically stable ” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in ominimal expansions of real closed fields. 1 Introduction and
Galois Groups of First Order Theories.
 Journal of Mathematical Logic
, 2000
"... We study the groups GalL (T ) and GalKP (T ), and the associated equivalence relations EL and EKP , attached to a first order theory T . An example is given where EL 6= EKP (a non Gcompact theory). It is proved that EKP is the composition of EL and the closure of EL . Other examples are given ..."
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Cited by 29 (9 self)
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We study the groups GalL (T ) and GalKP (T ), and the associated equivalence relations EL and EKP , attached to a first order theory T . An example is given where EL 6= EKP (a non Gcompact theory). It is proved that EKP is the composition of EL and the closure of EL . Other examples are given showing this is best possible. 1
Ominimal spectra, infinitesimal subgroups and cohomology
 J. of Symbolic Logic
, 2007
"... By recent work on some conjectures of Pillay, each definably compact group G in a saturated ominimal expansion of an ordered field has a normal “infinitesimal subgroup ” G 00 such that the quotient G/G 00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that t ..."
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Cited by 14 (5 self)
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By recent work on some conjectures of Pillay, each definably compact group G in a saturated ominimal expansion of an ordered field has a normal “infinitesimal subgroup ” G 00 such that the quotient G/G 00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ → G/G 00 sends exact sequences of definably compact groups into exacts sequences of Lie groups. We then study the connections between the Lie group G/G 00 and the ominimal spectrum ˜ G of G. We prove that G/G 00 is a topological quotient of ˜ G. We thus obtain a natural homomorphism Ψ ∗ from the cohomology of G/G 00 to the ( Čech)cohomology of ˜ G. We show that if G 00 satisfies a suitable contractibility conjecture then ˜ G 00 is acyclic in Čech cohomology and Ψ ∗ is an isomorphism. Finally we prove the conjecture in some special cases. 1
Model theoretic connected components of groups
, 2009
"... We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah from [21]. We consider also in this ..."
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Cited by 11 (2 self)
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We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah from [21]. We consider also in this context the multiplicative and the additive groups of some rings (including infinite fields).
Gcompactness and groups
, 2007
"... Abstract. Lascar described EKP as a composition of EL and the topological closure of EL ([1]). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a nonGcompact theory, we consider the following example. Assume G is a group ..."
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Cited by 8 (2 self)
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Abstract. Lascar described EKP as a composition of EL and the topological closure of EL ([1]). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a nonGcompact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M ′ consisting of M and X as two sorts, where X is an affine copy of G and in M ′ we have the structure of M and the action of G on X. We prove that the Lascar group of M ′ is a semidirect product of the Lascar group of M and G/GL. We discuss the relationship between Gcompactness of M and M ′. This example may yield new examples of nonGcompact theories. 1.
A note on groups definable in difference fields
 Proceedings of the AMS, 130 (2001), 205
, 2000
"... In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACF A is virtually definably embeddable in an algebraic group. We give a positive answer using routine ..."
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Cited by 7 (1 self)
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In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACF A is virtually definably embeddable in an algebraic group. We give a positive answer using routine
Weak forms of elimination of imaginaries
 Math. Logic Quart
"... We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl(A) and clo ..."
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Cited by 6 (0 self)
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We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl(A) and closed subgroups of the Galois group Aut(acl(A)/A). We also characterize when the topology of the Galois group is the quotient topology. 1
Introduction to the Lascar group
 in London Mathematical Society Lecture Notes Series
, 2002
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Borel equivalence relations and Lascar strong types
, 2013
"... The “space ” of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objec ..."
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Cited by 5 (3 self)
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The “space ” of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group GalL(T) of T, have welldefined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly “smooth iff trivial” The possibility of a descriptive settheoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [3], where the first example of a “non Gcompact theory ” was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [4], [5] and the generalizations in [7]. 0