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One-basedness and groups of the form G/G 00

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Abstract

We initiate a geometric stability study of groups of the form G/G 00, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M′ with a predicate for G 00 and check 1-basedness or non-1-basedness for G/G 00, where G is an additive truncation of M, a multiplicative truncation of M, SO 2(M) or one of its truncations; such groups G/G 00 are now interpretable in M′. We prove that the only 1-based groups are those where G is a sufficiently “big” multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.

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References

  1. Baisalov J., Poizat B.: Paires de structures o-minimales. J. Symb. Log. 63(2), 570–578 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hasson A., Onshuus A.: Embedded o-minimal structures. Bull. Lond. Math. Soc. 42(1), 64–74 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hrushovski E., Peterzil Y., Pillay A.: Groups, measures and the NIP. J. Am. Math. Soc. 21(2), 563–596 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hrushovski, E., Pillay, A.: On NIP and invariant measures. Preprint, arXiv:0710.2330 (2007)

    Google Scholar 

  5. Kuhlmann, S.: Ordered Exponential Fields, Fields Institute Monographs (2000)

  6. Loveys J., Peterzil Y.: Linear o-minimal structures. Israel J. Math. 81, 1–30 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Macpherson D., Marker D., Steinhorn C.: Weakly o-minimal structures and real closed fields. Trans. Am. Math. Soc. 352(12), 5435–5483 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Madden J., Stanton C.: One-dimensional Nash groups. Pac. J. Math. 154(2), 331–344 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Mellor T.: Imaginaries in real closed valued fields. Ann. Pure Appl. Log. 139(1-3), 230–279 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Penazzi, D.: Hyperdefinable groups and modularity. PhD. thesis, University of Leeds (2011)

  11. Peterzil Y., Starchenko S.: A trichotomy theorem for o-minimal structures. Proc. Lond. Math. Soc. 77(3), 481–523 (1998)

    Article  MathSciNet  Google Scholar 

  12. Pillay A.: On groups and fields definable in o-minimal structures. J. Pure Appl. Algebra 53(3), 239–255 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Pillay A.: Type-definability, compact lie groups, and o-Minimality. J. Math. Log. 4(2), 147–162 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pillay, A.: Canonical bases in o-minimal and related structures, preprint, Citeseer (2006)

  15. Ramakrishnan, J.: Definable linear orders definably embed into lexicographic orders in o-minimal structures, Preprint, arXiv:1003.5400 (2010)

  16. Razenj V.: One-dimensional groups over an o-minimal structure. Ann. Pure Appl. Log. 53(3), 269–277 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shelah S.: Dependent first order theories, continued. Israel J. Math. 173(1), 1–60 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tressl M.: Model completeness of o-minimal structures expanded by dedekind cuts. J. Symb. Log. 70(1), 29–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. van der Dries, L.: Tame topology and o-minimal structures. Lecture notes series, Cambridge university press, London Mathematical Society (1998)

Download references

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Correspondence to Davide Penazzi.

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Research supported by EPSRC grant EP/F009712/1.

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Penazzi, D. One-basedness and groups of the form G/G 00 . Arch. Math. Logic 50, 743–758 (2011). https://doi.org/10.1007/s00153-011-0246-7

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