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Topological dynamics and definable groups

Published online by Cambridge University Press:  12 March 2014

Anand Pillay
Anand Pillay*
Affiliation:
School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK, E-mail:[email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 657 - 666
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[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.CrossRefGoogle Scholar
[2]Chernikov, A. and Kaplan, I., Forking and dividing in NTP2 theories, this Journal, vol. 77 (2012), pp. 1–20.Google Scholar
[3]Ellis, R., Lectures on topological dynamics, Benjamin, 1969.Google Scholar
[4]Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), pp. 563–596.CrossRefGoogle Scholar
[5]Hrushovski, E. and Pillay, A., On NIP and invariant measures, Journal of the European Mathematical Society, vol. 13 (2011), pp. 1005–1061.Google Scholar
[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. 599–605.CrossRefGoogle Scholar
[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. 2341–2366.CrossRefGoogle Scholar
[8]Newelski, L., Topological dynamics of definable group actions, this Journal, vol. 74 (2009), pp. 50–72.Google Scholar
[9]Newelski, L., Model theoretic aspects of the Ellis semigroup, Israel Journal of Mathematics, vol. 190 (2012), pp. 477–507.CrossRefGoogle Scholar
[10]Pillay, A., Geometric stability theory, Oxford University Press, 1996.CrossRefGoogle Scholar
[11]Pillay, A., Type-definability, compact Lie groups and o-minimality, Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 147–162.CrossRefGoogle Scholar