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Generalised Bohr compactification and model-theoretic connected components

Published online by Cambridge University Press:  05 December 2016

KRZYSZTOF KRUPIŃSKI  and
ANAND PILLAY
KRZYSZTOF KRUPIŃSKI
Affiliation:
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. e-mail: [email protected]
ANAND PILLAY
Affiliation:
Department of Mathematics, University of Notre Dame, 281 Hurley Hall, Notre Dame, IN 46556, U.S.A. e-mail: [email protected]
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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*)000M 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*)000M 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*)000M 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 219 - 249
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Auslander, J. Minimal flows and their extensions. Mathematics Studies 153 (North-Holland, The Netherlands, 1988).Google Scholar
[2] Chernikov, A., Pillay, A. and Simon, P. External definability and groups in NIP theories. J. London Math. Soc. 90 (2014), 213–240.Google Scholar
[3] Chernikov, A. and Simon, P. Model theoretic tame dynamics, preprint.Google Scholar
[4] Conversano, A. and Pillay, A. Connected components of definable groups and o-minimality I. Adv. Math. 231 (2012), 605–623.Google Scholar
[5] Conversano, A. and Pillay, A. Connected components of definable groups and o-minimality II. Ann. Pure Appl. Logic. 66 (2015), 836–849.Google Scholar
[6] Gismatullin, J. Model theoretic connected components of groups. Israel J. Math. 184 (2011), 251–274.Google Scholar
[7] Gismatullin, J. and Krupiński, K. On model-theoretic connected components in some group extensions. J. Math. Log. 15 (2015), 1550009 (51 pages).Google Scholar
[8] Gismatullin, J. and Newelski, L. G-compactness and groups. Arch. Math. Logic. 47 (2008), 479–501.Google Scholar
[9] Gismatullin, J., Penazzi, D. and Pillay, A. On compactifications and the topological dynamics of definable groups. Ann. Pure Appl. Logic. 165 (2014), 552–562.Google Scholar
[10] Gismatullin, J., Penazzi, D. and Pillay, A. Some model theory of SL(2;ℝ). Fund. Math. 229 (2015), 117–128.Google Scholar
[11] Glasner, S. Proximal Flows Lecture Notes in Math. 517 (Springer, Germany, 1976).CrossRefGoogle Scholar
[12] Hrushovski, E. and Pillay, A. On NIP and invariant measures. J. Eur. Math. Soc. 13 (2011), 1005–1061.Google Scholar
[13] Jagiella, G. Definable topological dynamics and real Lie groups. Math. Log. Q. 61 (2015), 45–55.CrossRefGoogle Scholar
[14] Kaplan, I., Miller, B. and Simon, P. The Borel cardinality of Lascar strong types. J. London Math. Soc. 90 (2014), 609–630.Google Scholar
[15] Krupiński, K., Pillay, A. and Solecki, S.. Borel equivalence relations and Lascar strong types. J. Math. Log. 13 (2013), 1350008 (37 pages).Google Scholar
[16] Krupiński, K. and Rzepecki, T.. Smoothness of bounded invariant equivalence relations. J. Symb. Log. 81 (2016), 326–356.Google Scholar
[17] Newelski, L. Topological dynamics of definable group actions. J. Symb. Log. 74 (2009), 50–72.Google Scholar
[18] Newelski, L. Model theoretic aspects of the Ellis semigroup. Israel J. Math. 190 (2012), 477–507.CrossRefGoogle Scholar
[19] Peterzil, Y. Pillay's conjecture and its solution – a survey. In: Logic Colloquium 2007, 177–203 Lect. Notes Log. 35 (Assoc. Symbol. Logic, La Jolla, CA, 2010).Google Scholar