Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T05:11:10.778Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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