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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 16 (7 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
Connected components of definable groups and ominimality II, in preparation
"... In this sequel to [3] we try to give a comprehensive account of the “connected components ” G 00 and G 000 as well as the various quotients G/G 00, G/G 000, G 00 /G 000, for G a group definable in a (saturated) ominimal expansion of a real closed field. Key themes are the structure of G 00 /G 000 a ..."
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Cited by 3 (2 self)
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In this sequel to [3] we try to give a comprehensive account of the “connected components ” G 00 and G 000 as well as the various quotients G/G 00, G/G 000, G 00 /G 000, for G a group definable in a (saturated) ominimal expansion of a real closed field. Key themes are the structure of G 00 /G 000 and the problem of “exactness ” of the G ↦ → G 00 functor. We prove that the examples produced in [3] are typical, and that for any G, G 00 /G 000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup (where we allow the trivial Lie group). 1 Introduction and
Connected components of definable groups and ominimality
"... We give examples of definable groups G (in a saturated model, sometimes ominimal) such that G 00 = G 000, yielding also new examples of “non Gcompact ” theories. We also prove that for G definable in a (saturated) ominimal structure, G has a “bounded orbit ” (i.e. there is a type of G whose stab ..."
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Cited by 3 (2 self)
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We give examples of definable groups G (in a saturated model, sometimes ominimal) such that G 00 = G 000, yielding also new examples of “non Gcompact ” theories. We also prove that for G definable in a (saturated) ominimal structure, G has a “bounded orbit ” (i.e. there is a type of G whose stabilizer has bounded index) if and only if G is definably amenable, giving a positive answer to a conjecture of Newelski and Petrykowski in this special case of groups definable in ominimal structures. We also introduce and discuss further conjectures on bounded orbits in definable groups. These results and analyses are informed by a decomposition theorem for groups in ominimal structures. 1 Introduction and
EQUIVARIANT HOMOTOPY OF DEFINABLE GROUPS
, 905
"... Abstract. We consider groups definable in an ominimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is defin ..."
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Cited by 1 (0 self)
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Abstract. We consider groups definable in an ominimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the “compact domination conjecture ” proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to prove the following result. Let G and G ′ be two definably compact definably connected groups with isomorphic associated Lie groups. Then G and G ′ are definably homotopy equivalent. Moreover given a finite subgroup Γ of G, there is a definable homotopy equivalence f: G → G ′ that restricted to Γ is an isomorphism onto its image and such that f(cx) = f(c)f(x) for all c ∈ Γ and x ∈ G. In the semisimple case a stronger result holds: any Lie isomorphism from G/G 00 to G ′ /G ′00 induces a definable isomorphism from G to G ′. 1.
On the ominimal LScategory
, 2009
"... ABSTRACT. We introduce the ominimal LScategory of definable sets in ominimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the ominimal LScategory of definable groups. Along the way, we show that two definably connected defin ..."
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Cited by 1 (0 self)
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ABSTRACT. We introduce the ominimal LScategory of definable sets in ominimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the ominimal LScategory of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture. 1
Invariance results for definable extensions of groups
, 2010
"... We show that in an ominimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in ominimal expansions of the ordered ..."
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We show that in an ominimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in ominimal expansions of the ordered set of real numbers. With partial support from the FCT (Fundação para a Ciência e Tecnologia) program
Commutators in groups definable in ominimal structures
, 2010
"... We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups wit ..."
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We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in ominimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
Abstract
, 2007
"... 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 ..."
Abstract
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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) “generic compact domination ” for groups with finitely satisfiable generics, (vi) (a proof of) the compact domination conjecture in the ominimal case (i.e. for definably compact commutative groups in ominimal expansions of real closed fields) 1 Introduction and
Contents
"... Abstract. There are strong analogies between groups definable in ominimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a uniqu ..."
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Abstract. There are strong analogies between groups definable in ominimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsionfree subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsionfree subgroup H such that G/N can be decomposed as G/N = KH, with K ∩ H = {e}. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lielike decomposition of G where the role of maximal tori is played by maximal 0subgroups.