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Degrees of rigidity for Souslin trees

Published online by Cambridge University Press:  12 March 2014

Gunter Fuchs  and
Joel David Hamkins
Gunter Fuchs
Affiliation:
Westfälische Wilhelms-Universität Münster, Institut für Mathematische Logik und Grundlagenforschung, Einsteinstraße 62, 48149 Münster, Germany, E-mail: [email protected] Mathematics, The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, Ny 10016, USA
Joel David Hamkins
Affiliation:
Mathematics, The Graduate Center of the City University of New York, 365 Fifth Avenue, New York, Ny 10016, USA Mathematics, The College of Staten Island of Cuny, Staten Island, Ny 10314, USA, E-mail: [email protected], URL: http://jdh.hamkins.org
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Abstract

We investigate various strong notions of rigidity for Souslin trees, separating them under ⟡ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ⟡ that there is a group whose automorphism tower is highly malleable by forcing.

Keywords

Rigid Souslin treesdiamondautomorphism tower
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 2 , June 2009 , pp. 423 - 454
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Abraham, Uri, Construction of a rigid Aronszajn tree, Proceedings of the American Mathematical Society, vol. 77 (1979), no. 1, pp. 136–137.CrossRefGoogle Scholar
[2]Abraham, Uri and Shelah, Saharon, Isomorphism types of Aronszajn trees, Israel Journal of Mathematics, vol. 50 (1985), pp. 75–113.CrossRefGoogle Scholar
[3]Devlin, Keith J. and Johnsbråten, Havard, The Souslin problem, Lecture Notes in Mathematics, no. 405, Springer, Berlin, 1974.CrossRefGoogle Scholar
[4]Fuchs, Gunter and Hamkins, Joel David, Changing the heights of automorphism towers by forcing with Souslin trees over L, this Journal, vol. 73 (2008), no. 2, pp. 614–633.Google Scholar
[5]Gaifman, Haim and Specker, E. P., Isomorphism types of trees, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 1–6.CrossRefGoogle Scholar
[6]Hamkins, Joel David, Every group has a terminating transfinite automorphism tower, Proceedings of the American Mathematical Society, vol. 126 (1998), no. 11, pp. 3223–3226.CrossRefGoogle Scholar
[7]Hamkins, Joel David, How tall is the automorphism tower of a group?, Logic and algebra, AMS Contemporary Mathematics Series, vol. 302, 2001, pp. 49–57.Google Scholar
[8]Hamkins, Joel David and Thomas, Simon, Changing the heights of automorphism towers, Annals of Pure and Applied Logic, vol. 102 (2000), no. 1-2, pp. 139–157.CrossRefGoogle Scholar
[9]Hulse, J. A., Automorphism towers of polycyclic groups, Journal of Algebra, vol. 16 (1970), pp. 347–398.CrossRefGoogle Scholar
[10]Jech, Thomas, Automorphisms of ω1-trees, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 57–70.Google Scholar
[11]Jech, Thomas, Forcing with trees and ordinal definability, Annals of Mathematical Logic, vol. 7 (1974), pp. 387–409.CrossRefGoogle Scholar
[12]Jech, Thomas, Set theory, 3rd ed., Springer Monographs in Mathematics, 2003.Google Scholar
[13]Jech, Tomáš, Non-provability of Souslin's hypothesis, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 291–305.Google Scholar
[14]Kurepa, G., Ensembles ordonnées et ramifiés, Publications de l'Institut Mathématique, Univ. Belgrade, vol. 4 (1935), pp. 1–138.Google Scholar
[15]Rae, Andrew and Roseblade, James E., Automorphism towers of extremal groups, Mathematische Zeitschrift, vol. 117 (1970), pp. 70–75.CrossRefGoogle Scholar
[16]Scharfenberger-Fabian, Gido, Subalgebras of small Souslin algebras and maximal chains in Souslin algebras, Dissertation, Freie Universität Berlin, 03 2008.Google Scholar
[17]Tennenbaum, S., Souslin's problem. Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 60–63.CrossRefGoogle ScholarPubMed
[18]Thomas, Simon, The automorphism tower problem, Proceedings of the American Mathematical Society, vol. 95 (1985), pp. 166–168.CrossRefGoogle Scholar
[19]Thomas, Simon, The automorphism tower problem II, Israel Journal of Mathematics, vol. 103 (1998), pp. 93–109.CrossRefGoogle Scholar
[20]Thomas, Simon, The automorphism tower problem, to appear.Google Scholar
[21]Todorčević, Stevo, Rigid Aronszajn trees, Publications de l'Institut Mathematique (Beograd), Nouvelle Série, vol. 41 (1980), no. 27, pp. 259–265.Google Scholar
[22]Wielandt, H., Eine Verallgemeinerung der invarianten Untergruppen, Mathematische Zeitschrift, vol. 45 (1939), pp. 209–244.CrossRefGoogle Scholar