G-compactness and groups

We prove that for a finitely generated infinite nilpotent group G with structure (G, ·, . . .), the connected component G * 0 of a sufficiently saturated extension G * of G exists and equals n∈N {g n : g ∈ G * }. We construct an expansion of Z by a predicate (Z, +, P ) such that the type-connected component Z * 00 ∅ is strictly smaller than Z * 0 . We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

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“…(4) For n ≥ 2, if P is n-thick, then P is right and left (n − 1)-generic. (5) If P is right [resp. left] n-generic, then P −1 P [resp.…”

Section: Lemma 22mentioning

confidence: 99%

Model theoretic connected components of finitely generated nilpotent groups

2013

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“…Since a well-known result (see e.g. [8,Proposition 3.5]) says that G * 00 A /G * 000 A is the closure of the neutral element in G * /G * 000 A , we conclude that in order to finish the proof, we need to show that f [S] is closed in G * /G * 000 A . All the sets Y , S and P u are defined in terms of the idempotent u ∈ M chosen at the beginning.…”

Section: The Proof Of Theorem 02mentioning

confidence: 95%

“…see [12,8,6]). In particular, they play a fundamental role in the study of stable, simple and NIP groups, and they are precisely related to strong types in different senses [8]. Note also that all quotients such as G * /G * 0 A , G * /G * 00 A , G * /G * 000 A or G * 00 A /G * 00 A are certain invariants of the group G (in the sense that they do not depend on the choice of the monster model) and it is desirable to understand their algebraic, topological and possibly some other structure.…”

Section: Introductionmentioning

confidence: 99%

Generalised Bohr compactification and model-theoretic connected components

Krupiński¹,

Pillay²

2016

Math. Proc. Camb. Phil. Soc.

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“…Fixing a point x 0 ∈ X, the isomorphism F say between Aut(N * ) and G * ⋊ Aut(M * ) takes f ∈ Aut(N * ) to (g, f |M * ), where f (x 0 ) = g −1 · x 0 . In [7], it is observed (Proposition 3.3 there) that F induces an isomorphism between Gal L (T h(N)) and (G * /(G * ) 000 ) ⋊ Gal L (T h(M)), as well as between Gal KP (T h(N)) and (G * /(G * * ) 00 ) ⋊ Gal KP (T h(M)). One deduces an isomorphism between Gal 0 (T h(N)) and (G * ) 00 /(G * ) 000 ⋊ Gal 0 (T h(M)).…”

Section: The Lascar Groupmentioning

confidence: 99%

Borel Equivalence Relations and Lascar Strong Types

2013

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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 Gal L (T ) of T , have well-defined 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 set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [2], where the first example of a "non G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [3], [4] and the generalizations in [6].

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G-compactness and groups

Cited by 20 publications

References 7 publications

Model theoretic connected components of finitely generated nilpotent groups

Model theoretic connected components of finitely generated nilpotent groups

Generalised Bohr compactification and model-theoretic connected components

Borel Equivalence Relations and Lascar Strong Types

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