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Automorphism groups of trivial strongly minimal structures

Published online by Cambridge University Press:  12 March 2014

Thomas Blossier
Thomas Blossier*
Affiliation:
Ufr de Mathématiques, Université Lyon I, Bâtiment Doyen Jean Braconnier, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, France, E-mail: [email protected]
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Abstract

We study automorphism groups of trivial strongly minimal structures. First we give a characterization of structures of bounded valency through their groups of automorphisms. Then we characterize the triplets of groups which can be realized as the automorphism group of a non algebraic component, the subgroup stabilizer of a point and the subgroup of strong automorphisms in a trivial strongly minimal structure, and also we give a reconstruction result. Finally, using HNN extensions we show that any profinite group can be realized as the stabilizer of a point in a strongly minimal structure of bounded valency.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 644 - 668
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[Blo]Blossier, T., Ensembles minimaux localement modulaires, ThÈse de doctoral, UniversitÉ Paris 7, 2001.Google Scholar
[Coh]Cohen, D. E., Combinatorial group theory: a topological approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
[E-I-M]Evans, D. M., Ivanov, A. A., and Macpherson, D., Finite covers, Model theory of groups and automorphism groups (Blaubeuren, August 1995), Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
[Hig]Higgins, P. J., Introduction to topological groups, London Mathematical Society Lecture Notes 152, Cambridge, 1974.Google Scholar
[Hru 92]Hrushovski, E., Unimodular minimal structures, The Journal of the London Mathematical Society, vol. 46 (1992), no. 3, pp. 385–396.CrossRefGoogle Scholar
[Hru 94]Hrushovski, E., Finitely axiomatizable ℵ1 categorical theories, this Journal, vol. 59 (1994), no. 3, pp. 838–844.Google Scholar
[Iva 89]Ivanov, A. A., The problem of finite axiomatizability for strongly minimal theories of graphs, Algebra and Logic, vol. 28 (1989), pp. 183–194.CrossRefGoogle Scholar
[Iva 93]Ivanov, A. A., Strongly minimal structures with disintegrated algebraic closure and structures of bounded valency, Proceedings of the Tenth Easter Conference on Model Theory (Weese, M. and Wolter, H., editors), 1993.Google Scholar
[Nek]Nekrashevych, V., Stabilizers of transitive actions on locally finite graphs, International Journal of Algebra and Computation, vol. 10 (2000), no. 5, pp. 591–602.CrossRefGoogle Scholar
[Pil]Pillay, A., Geometric stability theory, Oxford University Press, 1996.CrossRefGoogle Scholar