Definable groups and compact <i>p</i>
-adic Lie groups

Onshuus, Alf; Pillay, Anand

doi:10.1112/jlms/jdn018

Cited by 32 publications

(49 citation statements)

References 11 publications

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“…In fact the latter proved in addition that G/G 00 is a compact Lie group. For groups definable over Q p in a model of T h((Q p ) an ) this was done in [16]. For groups definable in Pressburger Arithmetics, it follows from work of Onshuus [15].…”

Section: Proposition 55 Conditions (1) and (2) Are Equivalent (3) mentioning

confidence: 99%

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Groups, measures, and the NIP

Hrushovski

Peterzil

Pillay

2007

J. Amer. Math. Soc.

185

354

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We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups G in saturated o-minimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant finitely additive probability measure on definable subsets of G. We finally introduce a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the o-minimal case.

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Section: Proposition 55 Conditions (1) and (2) Are Equivalent (3) mentioning

confidence: 99%

“…Secondly (and more substantially) the above count of torsion points by Edmundo and Otero was only carried out for expansions of real closed fields. The conjecture was proved separately for groups definable in ordered vector spaces over division rings (see [15], [8]). …”

Section: Claimmentioning

confidence: 99%

Groups, measures, and the NIP

Hrushovski

Peterzil

Pillay

2007

J. Amer. Math. Soc.

185

354

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“…From the analysis there, one can, with additional work, obtain a characterization of the definable groups G with fsg, at least when G is abelian: they are precisely the (abelian) groups G such that there is a definable homomorphism h : G → A where A is internal to the value group and is definably compact, and Ker(h) is stably dominated. The p-adically closed field case is considered in [24], where among other things it is pointed out that definably compact groups defined over the standard model have fsg.…”

Section: Generics and Forking In Definable Groupsmentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

151

269

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Abstract. 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 [16]. 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 o-minimal expansions of real closed fields.

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“…Suppose now that

C_{K}

is a p ‐adically closed field, then one can find a description of definable (and type‐definable) subgroups of

E (C_{K})

in [, Chapter 7] and . Even though in this case the theory does not admit elimination of imaginaries, since we obtained the isomorphism explicitly, we get that

gal (L / K)

{Gal}_{frakturU} false(L / K false)

is isomorphic to the

C_{K}

‐points or to its

C_{U}

‐points, respectively, of a group

{scriptL}_{rings} false(C_{K} false)

‐definable.…”

Section: Examples Of Strongly Normal Extensionsmentioning

confidence: 99%

On differential Galois groups of strongly normal extensions

Brouette

Point

2018

Mathematical Logic Qtrly

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We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in sans-serifCODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.

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Definable groups and compact p -adic Lie groups

Cited by 32 publications

References 11 publications

Groups, measures, and the NIP

Groups, measures, and the NIP

On NIP and invariant measures

On differential Galois groups of strongly normal extensions

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