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  1. Borel Equivalence Relations and Lascar Strong Types.Krzysztof Krupiński, Anand Pillay & Sławomir Solecki - 2013 - Journal of Mathematical Logic 13 (2):1350008.
    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 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 of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...)
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  • On Model-Theoretic Connected Components in Some Group Extensions.Jakub Gismatullin & Krzysztof Krupiński - 2015 - Journal of Mathematical Logic 15 (2):1550009.
    We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new (...)
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  • Connected Components of Definable Groups, and o-Minimality II.Annalisa Conversano & Anand Pillay - 2015 - Annals of Pure and Applied Logic 166 (7-8):836-849.
  • Model Theoretic Connected Components of Finitely Generated Nilpotent Groups.Nathan Bowler, Cong Chen & Jakub Gismatullin - 2013 - Journal of Symbolic Logic 78 (1):245-259.
    We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for (...)
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  • Smoothness of Bounded Invariant Equivalence Relations.Krzysztof Krupiński & Tomasz Rzepecki - 2016 - Journal of Symbolic Logic 81 (1):326-356.