Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T07:42:50.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups definable in linear o-minimal structures: the non-compact case

Published online by Cambridge University Press:  12 March 2014

Pantelis E. Eleftheriou
Pantelis E. Eleftheriou*
Affiliation:
CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal, E-mail: pelefthe@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Let = ⟨M, +, <, 0, S⟩ be a linear o-minimal expansion of an ordered group, and G = ⟨G, ⊕,eG ) an n-dimensional group definable in . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L. for some convex ∨-definable subgroup U of ⟨Mn , +⟩ and a lattice L of rank equal to the dimension of the ‘compact part’ of G.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 208 - 220
Copyright
Copyright © Association for Symbolic Logic 2010

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

[BOPP] Berarducci, A., Otero, M., Peterzil, Y., and Pillay, A., A descending chain condition for groups definable in o-minimal structures, Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303–313.Google Scholar
[Bour] Bourbaki, N., Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics, Springer-Verlag, Berlin, 1989.Google Scholar
[EdEl] Edmundo, M. and Eleftheriou, P., Definable group extensions in semi-bounded o-minimal structures, Mathematical Logic Quarterly, vol. 55 (2009), pp. 598–604.Google Scholar
[ElSt] Eleftheriou, P. and Starchenko, S., Groups definable in ordered vector spaces over ordered division rings, this Journal, vol. 72 (2007), pp. 1108–1140.Google Scholar
[HPP] Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), pp. 563–596.Google Scholar
[LP] Loveys, J. and Peterzil, Y., Linear o-minimal structures, Israel Journal of Mathematics, vol. 81 (1993), pp. 1–30.Google Scholar
[Ons] Onshuus, A., Groups definable in ⟨ℚ, +, <˩, Preprint, 2005.Google Scholar
[PePi] Peterzil, Y. and Pillay, A., Generic sets in definably compact groups, Fundamenta Mathematical vol. 193 (2007), no. 2, pp. 153–170.Google Scholar
[PeSt] Peterzil, Y. and Starchenko, S., Definable homomorphisms of abelian groups in o-minimal structures, Annals of Pure and Applied Logic, vol. 101 (2000), pp. 1–27.Google Scholar
[PeS] Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society. Second Series, vol. 69 (1999), pp. 769–786.Google Scholar
[Pil] Pillay, A., On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.Google Scholar
[Pi2] Pillay, A., Type-definability, compact Lie groups, and o-minimality, Journal of Mathematical Logic, vol. 4 (2004), pp. 147–162.Google Scholar
[Str] Strzebonski, A., Euler characteristic in semialgebraic and other o-minimal groups, Journal of Pure and Applied Algebra, vol. 96 (1994), pp. 173–201.Google Scholar