Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T12:59:01.736Z Has data issue: false hasContentIssue false
  • English
  • Français

THE ELLIS GROUP CONJECTURE AND VARIANTS OF DEFINABLE AMENABILITY

Published online by Cambridge University Press:  21 December 2018

GRZEGORZ JAGIELLA
GRZEGORZ JAGIELLA*
Affiliation:
INSTYTUT MATEMATYCZNY UNIWERSYTETU WROCŁAWSKIEGOPL. GRUNWALDZKI 2/4 50-384WROCŁAW, POLANDE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider definable topological dynamics for NIP groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable flow. This description shows that the Ellis group is of bounded size. Under additional assumptions, it is shown to be independent of the model, proving a conjecture proposed by Newelski. Finally we apply the results to new classes of groups definable in o-minimal structures, generalizing all of the previous results for this setting.

Keywords

Primary 03C9954H20Secondary 03C64definable topological dynamicsdefinable amenabilityo-minimality
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1376 - 1390
Copyright
Copyright © The Association for Symbolic Logic 2018 

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

Chernikov, A., Pillay, A., and Simon, P., External definability and groups in NIP theories. Journal of the London Mathematical Society, vol. 90 (2014), no. 1, pp. 213–240.CrossRefGoogle Scholar
Chernikov, A. and Simon, P., Definably amenable NIP groups. Journal of the American Mathematical Society, vol. 31 (2018), no. 3, pp. 609–641.CrossRefGoogle Scholar
Conversano, A., Maximal compact subgroups in the o-minimal setting. Journal of Mathematical Logic, vol. 13 (2013), no. 1, pp. 1–15.CrossRefGoogle Scholar
Conversano, A., A reduction to the compact case for groups definable in o-minimal structures, this Journal, vol. 79 (2014), no. 1, pp. 45–53.Google Scholar
Conversano, A. and Pillay, A., Connected components of definable groups and o-minimality I. Advances in Mathematics, vol. 231 (2012), no. 2, pp. 605–623.CrossRefGoogle Scholar
Gismatullin, J., Penazzi, D., and Pilay, A., Some model theory of $SL\left( {2,{\Cal R}} \right) $. Fundamenta Mathematicae, vol. 229 (2014), pp. 117–128.CrossRefGoogle Scholar
Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563–596.CrossRefGoogle Scholar
Hrushovski, E. and Pillay, A., On NIP and invariant measures. Journal of the European Mathematical Society, vol. 13 (2011), no. 4, pp. 1005–1061.CrossRefGoogle Scholar
Jagiella, G., Definable topological dynamics and real Lie groups. Mathematical Logic Quarterly, vol. 61 (2015), no. 1–2, pp. 45–55.CrossRefGoogle Scholar
Newelski, L., Topological dynamics of definable group actions, this Journal, vol. 74 (2009), no. 1, pp. 50–72.Google Scholar
Newelski, L., Model theoretic aspects of the Ellis semigroup. Israel Journal of Mathematics, vol. 190 (2012), no. 1, pp. 477–507.CrossRefGoogle Scholar
Newelski, L., Topological dynamics of stable groups, this Journal, vol. 79 (2014), no. 4, pp. 1199–1223.Google Scholar
Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups. Journal of the London Mathematical Society, vol. 59 (1999), no. 3, pp. 769–786.CrossRefGoogle Scholar
Pillay, A., Topological dynamics and definable groups, this Journal, vol. 78 (2013), no. 2, pp. 657–666.Google Scholar
Pillay, A. and Yao, N., On minimal flows, definably amenable groups, and o-minimality. Advances in Mathematics, vol. 290 (2016), pp. 483–502.CrossRefGoogle Scholar
Shelah, S., Dependent first order theories, continued. Israel Journal of Mathematics, vol. 173 (2009), no. 1, p. 1.CrossRefGoogle Scholar
Simon, P., A Guide to NIP Theories, Lecture Notes in Logic, vol. 44, Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Yao, N. and Long, D., Topological dynamics for groups definable in real closed field. Annals of Pure and Applied Logic, vol. 166 (2015), no. 3, pp. 261–273.CrossRefGoogle Scholar