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Generalised Bohr compactification and model-theoretic connected components

  • KRZYSZTOF KRUPIŃSKI (a1) and ANAND PILLAY (a2)
Abstract

For a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.

Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000 M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000 M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG (M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000 M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.

In the situation when all types in SG (M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.

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[1] Auslander, J. Minimal flows and their extensions. Mathematics Studies 153 (North-Holland, The Netherlands, 1988).
[2] Chernikov, A., Pillay, A. and Simon, P. External definability and groups in NIP theories. J. London Math. Soc. 90 (2014), 213240.
[3] Chernikov, A. and Simon, P. Model theoretic tame dynamics, preprint.
[4] Conversano, A. and Pillay, A. Connected components of definable groups and o-minimality I. Adv. Math. 231 (2012), 605623.
[5] Conversano, A. and Pillay, A. Connected components of definable groups and o-minimality II. Ann. Pure Appl. Logic. 66 (2015), 836849.
[6] Gismatullin, J. Model theoretic connected components of groups. Israel J. Math. 184 (2011), 251274.
[7] Gismatullin, J. and Krupiński, K. On model-theoretic connected components in some group extensions. J. Math. Log. 15 (2015), 1550009 (51 pages).
[8] Gismatullin, J. and Newelski, L. G-compactness and groups. Arch. Math. Logic. 47 (2008), 479501.
[9] Gismatullin, J., Penazzi, D. and Pillay, A. On compactifications and the topological dynamics of definable groups. Ann. Pure Appl. Logic. 165 (2014), 552562.
[10] Gismatullin, J., Penazzi, D. and Pillay, A. Some model theory of SL(2;ℝ). Fund. Math. 229 (2015), 117128.
[11] Glasner, S. Proximal Flows Lecture Notes in Math. 517 (Springer, Germany, 1976).
[12] Hrushovski, E. and Pillay, A. On NIP and invariant measures. J. Eur. Math. Soc. 13 (2011), 10051061.
[13] Jagiella, G. Definable topological dynamics and real Lie groups. Math. Log. Q. 61 (2015), 4555.
[14] Kaplan, I., Miller, B. and Simon, P. The Borel cardinality of Lascar strong types. J. London Math. Soc. 90 (2014), 609630.
[15] Krupiński, K., Pillay, A. and Solecki, S.. Borel equivalence relations and Lascar strong types. J. Math. Log. 13 (2013), 1350008 (37 pages).
[16] Krupiński, K. and Rzepecki, T.. Smoothness of bounded invariant equivalence relations. J. Symb. Log. 81 (2016), 326356.
[17] Newelski, L. Topological dynamics of definable group actions. J. Symb. Log. 74 (2009), 5072.
[18] Newelski, L. Model theoretic aspects of the Ellis semigroup. Israel J. Math. 190 (2012), 477507.
[19] Peterzil, Y. Pillay's conjecture and its solution – a survey. In: Logic Colloquium 2007, 177203 Lect. Notes Log. 35 (Assoc. Symbol. Logic, La Jolla, CA, 2010).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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