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Topological dynamics and definable groups

  • Anand Pillay (a1)

Abstract

We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G(M) on its “external type space” SG.ext(M), can explain, account for, or give rise to, the quotient G/G00, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or f sg) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of G(M) for measure-stable groups. We also point out that for G definably amenable (in a NIP theory) G/G00 can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global f-generic types.

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[1]Akin, E., Auslander, J., and Glasner, E., The topological dynamics of Ellis actions. Memoirs of the American Mathematical Society, no. 913, American Mathematical Society, 2008.
[2]Chernikov, A. and Kaplan, I., Forking and dividing in NTP2 theories, this Journal, vol. 77 (2012), pp. 120.
[3]Ellis, R., Lectures on topological dynamics, Benjamin, 1969.
[4]Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), pp. 563596.
[5]Hrushovski, E. and Pillay, A., On NIP and invariant measures, Journal of the European Mathematical Society, vol. 13 (2011), pp. 10051061.
[6]Hrushovski, E., Pillay, A., and Simon, P., A note on generically stable measures and fsg groups, Notre Dame Journal of Formal Logic, vol. 53 (2012), pp. 599605.
[7]Hrushovski, E., Pillay, A., and Simon, P., On generically stable and smooth measures in NIP theories. Transactions of the American Mathematical Society, vol. 365 (2013), pp. 23412366.
[8]Newelski, L., Topological dynamics of definable group actions, this Journal, vol. 74 (2009), pp. 5072.
[9]Newelski, L., Model theoretic aspects of the Ellis semigroup, Israel Journal of Mathematics, vol. 190 (2012), pp. 477507.
[10]Pillay, A., Geometric stability theory, Oxford University Press, 1996.
[11]Pillay, A., Type-definability, compact Lie groups and o-minimality, Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 147162.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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